"Intelligence is a measure of your overall mental acuity, and affects the number of Experience Points earned."
The only thing Intelligence itself does is increase the bonus XP you earn. This isn't a bad thing, as improving the time spent-to-awesomeness attained ratio is always a good thing. Still, you can always go out and kill a little more, pick some more locks, build some more items, that sort of things. That being the case, it's hard to argue you should invest in Intelligence for the sake of its benefits. Getting rid of skill points really downgraded the importance of poor Intelligence. Anyway, onto the perks.
V.A.N.S. will create a handy-dandy trail for you to follow leading to the quest objective currently selected in your Pip-Boy. This might be nifty if you don't have an awesome guide to follow, but you clearly do. Still, it can help clear up ambiguity a bit.
Medic increases the effectiveness of your Stimpaks and RadAway, which isn't a bad thing, it's just that they stretch this benefit out over four perk ranks. The biggest boon is rank four's increased speed (not that the 100% healing offered isn't good), but think about it for a moment - Stimpaks and RadAway are common enough that on the default difficulty you should have more than enough to see you through the game. Heck, you should end up building up a massive surplus if you're careful. Also, by the time you can get rank four you'll need to hit level forty-nine, by which point you've had more than enough time to get Ballistic Weave Mk5. This won't eliminate damage, of course, but very few things should be able to damage you faster than you can heal, even with a slower rate. You do, however, need one rank in this perk to build Clinics in your settlements, so it might be worth investing a perk into Medic just for that.
Gun Nut is one of those rare perks in Fallout 4 which is almost mandatory. Even if you prefer energy weapons, some mods just require Gun Nut, and the superior Heavy Machinegun Turret does as well (along with other, less interesting defenses). The main draw of Gun Nut, however, is that it allows you to greatly improve the potency of your mundane weapons, from the humble Pipe Pistol to the Gauss Rifle. If there's one nit-pick to this perk (besides the fact that you can simply opt to use Energy Weapons instead) is the fact that you can technically find all the mods you can make yourself. If you find them, you can dismantle the guns that have them and add them onto other guns you prefer. Find a silenced 10mm Pistol? Take the silencer off and you can put it on a 10mm Pistol with a superior barrel. Of course, the drawback to this is that this brings luck of the drop into the mix, and almost ensures that enemies you fight will have superior weapons until you can kill them and take their guns. Spending perks on Gun Nut will ensure you've always got superior weaponry, and will honestly go a long way towards making the game laughably easy - almost as much as Armorer and Ballistic Weave Mk5 does! Still, if you like winning and staying ahead of foes, keeping up with Gun Nut is a great way to do it.
Like Locksmith, Hacker allows you to deal with various grades of pesky terminals in the wasteland. This typically allows you to open mag-locked doors that can't otherwise be opened (more than a few suits of Power Armor will be obtained this way), manipulated automated security, or, more mundanely, bypass conventional locks. Like Locksmith, bringing an NPC around with you (in this case, Nick Valentine) will allow you to use Nick's skills to handle all but [Master] terminals. Still, it's lame to be bogged down with an NPC because you're too lazy to invest three perks into Hacker, and it's even lamer to avoid terminals altogether. Make your life easier, drop three ranks in this, and become the hackster master. The fourth rank is bogus, as "getting locked out" of terminals just makes you wait a bit. Save your perk and have some patience. All in all, there are less terminals to hack in the Commonwealth than locks, so Hacking isn't quite as useful as Lock Pick.
Weapons can be broken down into Steel, Wood and Plastic on the outset, but if you invest in this perk (up to two ranks - three with the Far Harbor DLC) you can get many more materials, usually somewhat rarer ones. The amount of components you'll gain varies by weapon; you won't find Pipe Pistols any more rewarding with this perk, but energy weapons tend to go from breaking down into merely Plastic to adding Screws, Circuitry, Fiber Optics, and more. Also, Pipe Rifles will start breaking down into Copper, which is nice, and most high-end guns (Assault Rifles, Combat Rifles) turn into mounds of Aluminum. Also useful is the ability at rank two to tag specific components, which cause items and containers where said components can be found to glow green, might makes targeted harvesting much easier.
The biggest winners, as far as resources go, are uncommon - but useful - components like Aluminum, Copper, Circuitry and Screws. Be wary, however, that the amount of Steel you'll get at higher ranks of this perk will actually decrease on many weapons. If you craft or build a lot, this perk might be worth an investment, especially at rank three where you gain far more parts on some items that at previous ranks. You can also sell the excess parts in bulk for a nice proft but is all this necessary? It is more lucrative than simply selling the weapons and buying junk containing the parts you need? Not really. Again, if you break a bunch of stuff down and are willing to drastically over-level, this might be worthwhile, but for almost every player, you're not going to benefit from this perk much.
The Energy Weapons equivalent of Gun Nut, Science is required to mod every Energy Weapon, build a few items at settlements, and create a handful of other mods (pretty much every high-end scope). Simply put, if you like Energy Weapons, you're going to want this perk. There aren't really many weapons that fall under the Energy Weapons umbrella, but the few weapons that are there are supremely moddable. For example, your humble Laser Pistol can be turned into an automatic pistol, a sniper rifle, a shotgun, a non-auto rifle and an automatic rifle. Pretty much any weapon you could want. Energy Weapons proliferate a bit later than normal guns, and Science! typically has higher level requirements, but Energy Weapons are just as good as conventional guns, and for all the reasons you might want Gun Nut, you'll want Science! but perhaps not both.
Note: Science! is required to build some objects at the Robot Workbench, which is added with the Automatron DLC. Science! (Rank 4) is also required to build the incredibly handy "Generator - Fusion", which is added with the Wasteland Workshop DLC.
Each rank of Chemist (there are four in all) increases the duration of chems by 50%, up to a maximum of - use that math - 200%! If you like using chems, eh, you can always just use more. It's hard to imagine a situation when you'd need a chem to last longer, as one chem's duration should last an entire battle. Sure, there's radiation and Rad-X, but prolonged radiation exposure is rare, and you can always wear a Hazmat Suit or Power Armor instead. On the other hand, you do need one rank in this perk to create some of the more awesome chems in the game, like Overdrive and UltraJet, and they're powerful enough that the odd indulgence might be worth it. If that interests you, get one rank, if not, don't.
An interesting perk in Fallout 3 has been greatly reduced in value for Fallout 4. Having shed its brute damage bonus against robots this perk now allows you to sneak up on robots and hit the [Square] or [X] button to hack them, which allows you to shut them down or cause them to self-destruct (rank one), incite them to attack anything in sight (rank two) or give them more detailed commands (rank three). Fair enough, but the difficulty in sneaking up to robots - especially if they have allies nearby - makes this perk greatly unappealing for most situations. Lone robots are much easier to target, but it's also rather redundant, since they can usually be easily destroyed without this perk. Unless you really want a robot companion (aside from the ones already in the game), there's little reason to bother with this perk, and the novelty can't be worth three perks. Wait, you say, didn't you give "Wasteland Whisperer" a better score for essentially the novelty of controlling a Mirelurk Queen or Super Mutant Behemoth? Yes, but you can also use that perk to pacify any of the many, many, many Super Mutants, Mirelurks and Deathclaws roaming the wasteland, and you don't need to sneak up on anything to get it to work, making it far more useful than "Robotics Expert".
Note: The Automatron DLC adds the Robot Workbench, which allows you to craft various robots. Some high-end robot parts/mods require ranks in Robotics Expert.
Another perk that tries to lure you into wasting levels on it for the dubious benefit of retaining a few items that are easily found through exploration. The three ranks of Nuclear Physicist extend - to various degrees - the longevity of your Fusion Cores. It also increases the damage radiation weapons do, but since these are rare, and unimpressive (most conventional guns and Energy Weapons will kill things faster, Feral Ghouls and Robots are outright immune, and many irradiated critters are resistant), this isn't much of a bonus. Oh, and then there's the whole "use spent Fusion Cores like grenades" thing, which will happen so infrequently as to be a non-factor. All that taken into consideration, the only reason you should consider this perk is if you have some goal to remain in Power Armor the entire game (which is patently unnecessary thanks to Ballistic Weave), and even then, you can always buy, steal or find more Fusion Cores.
When you drop below 20% of your health you'll get a time slow effect (lasts for around ten seconds, once per activation) and will gain a variable amount of a damage boost (20%, 30% or 40%) and Damage Resistance (20, 30 or 40) as long as you remain severely injured. With a third rank in this you also gain health regen after every kill, which, if anything, will merely shorten your hulk period. 20% or less of your health is cutting it awfully close, but the damage boost is quite impressive, if you're willing to risk it. If you're been paying attention you should know that the Damage Resistance boost isn't terribly impressive, either. If you want a high-risk, damage boost kind of gameplay, this perk might interest you, but given all the downsides (high Intelligence requirement, low health threshold) it's hard to strong recommend it. And if you do want the boost for any significant length of time, make sure you don't get any health regen, or your stay in Nerd Rage won't be very long. As for a "saving grace" perk, since you can jump into your Pip-Boy at any time and use a Stimpak or another chem that slows down time, it shouldn't even be considered for that.