Everyone knows the Brotherhood runs around collecting tech throughout the wasteland, so they should have some pretty cool stuff. But now there's the Institute; an unknown quantity.
Will I miss out on some new weapons, armor or 'perks' by siding with either?
I've beaten the game with each of the Brotherhood of Steel and Institute multiple times. I can't claim to have done every single possible side quest for both of these two factions, but I've done most of them. I've supplemented my experience with information from Nukapedia, linked throughout the answer.
Summary: The Brotherhood of Steel gives you a lot of good stuff, mostly power armor pieces. The Institute gives you basically nothing.
Notably, nearly every one of these quests can be done before choosing a side. In fact, the only quest reward here you can't get if you side with the Institute in the end is a T60 Jumpjet, which you can build yourself. You won't lose any quest rewards by siding with the BoS, although you lose out on being able to buy Synth Relay Grenades after the game is over, and the opportunity to kill Maxson and take his stuff.
Let's first look at loot you get from quests that are required to finish the game with either of these factions.
The Institute gives you: nothing.
You literally don't get any items at all, unless you count a paint mod for X-01 armor that you don't get until the quest line is complete.
The brotherhood gives you:
Call to Arms gives you Righteous Authority. This is one of the best guns in the game, if not the best. You can easily get this gun within the first few levels and use it for the rest of the game. This quest occurs early, and you can do it regardless of which faction you end up supporting.
Shadow of Steel gives you a full suit of T-60b power armor. This suit of power armor is certainly not the best possible, but it's good enough. Depending on how quickly you finish the game, this could potentially be end game equipment as well. This quest also gives you Paladin Danse as a companion. Although not technically an item, Danse is an amazing asset who can single-handedly win a lot of fights.
Show No Mercy gives you Vertibird Signal Grenades. These can be used to call up a vertibird, complete with minigun. The vertibird will fly you to any previously discovered location. You can change course or order it to land anywhere along the way.
Liberty Reprimed gives you a T-60 Medic Pump, which as a T-60 power armor chest modification. It's not amazing, but if you haven't invested in Science! then you probably have a free slot to stick it in, especially with the free set of T-60 power armor you got earlier.
Blind Betrayal gives you a full set of T-60d power armor. This is better than the T-60b armor you already received, and has a nice paint job.
The next quests are Tactical Thinking, which causes you to become enemies with the Railroad, and Spoils of War, which causes you to become enemies with the Institute. So you can still get all the above loot, and then finish the game with the Institute.
There are a number of optional quests available for each faction. Here again, the BoS gives you stuff, and the Institute really doesn't.
The Insitute gives you:
There are also some radiant quests that give you common items like grenades or fusion cells.
None of these side quests require that you progress far enough to become enemies with the BoS.
The Brotherhood gives you:
The Lost Patrol gives you a Steadfast BOS Combat Armor Chest Piece, which is pretty great earlier in the game, and not too bad later on. It gives you extra damage resistance as your health gets lower. You can also get the Survivor's Special unique laser pistol.
Duty or Dishonor gives you the Exemplar's T-60c Torso, which is a great chest piece that reduces AP costs.
A Loose End gives you the Visionary's T-60c Helmet, which is another great power armor piece. This one increases your AP refresh speed.
The BoS radiant quests just give you caps and XP.
None of these side quests require that you progress far enough to become enemies with the Institute.
There are certain items that aren't given to the player as rewards, but are still only available during certain, faction-specific quests.
If you side with the Brotherhood, then you won't ever get the chance to kill Elder Maxson while he is wearing his suit of T-60f power armor with a unique Brotherhood of Steel Elder paint job, or while he is wielding his ultra-powerful unique weapon: Final Judgement. Both of these can be acquired by taking certain steps during Airship Down.
If you side with the Institute, then you can buy Synth Relay Grenades from the Institute vendor after the main quest. Also, once you become enemies with the Institute, any synths you summon will be hostile to you.
Not true you get grenades that spawn mark one synths, which are not that good but are fun to play with. At the very end when you being down the brotherhood of steel you get T V11 60F bos elder power armor set from killing elder maxon snitch is pretty hard with anything but the fat man, be care full and not lose his body, you also get the final judgement it's super good💦💦
You also forget about getting the cool Courser companion, and yeah the relay grenades are actually given to you during a quest. Plus, if you don't side with the Institute, the grenades are pointless. Also, end game you have a lot of synths roaming Commonwealth that can aid you in battle. You also get the best armor, Heavy Synth Armor plus a piece with the chameleon effect on it can be bought. No more legendary farming for it! And of course, the Waser Wifle.
I'm an Institute joiner, and while the above post is correct in that the Institute does not give much, there are some other positives.
1) INSTANT FAST TRAVEL to the Institute from any outdoor location. It doesn't take you anywhere else like the BOS vertibird, but there's also no waiting and it is in the middle of the map. It's pretty nice when you are overencumbered and want to drop stuff off. You can only fast travel out to the CIT Ruins right below the Institute though.
2) There is a doc in the med/bed bay in the Institute that cures you for free. Sometimes he is bugged at first but there's an easy one-time fix for console/PC both.
3) The Institute has a vendor with ~700 caps available for trading and 1 or 2 legendary synth armor or weapons for sale. There's also a food/water vendor and fresh water fountains in the Institute.
4) Tons of space in your room (at least 14 containers) to store your items.
5) A big DOWNSIDE is you cannot build anything in the Institute and there are NO CRAFTING STATIONS. I put my crafting stations in Hangman's Alley across the bridge.
6) Minor downside: all those viable blood samples and technical documents are useless if you join the Institute (I believe you can save them for BOS missions otherwise).
7) No offense to the BOS, but there are a lot of T-60 armors to find even without the BOS rewards out there. As a crafter, you can make all the same mods too.
8) You can buy unlimited synth grenades (~66-100 caps, I can't remember). Each one pops out a synth to fight your enemy. When the synth dies, you can loot them too.
I play on Survival so it is kind of a pain having to slow travel everywhere after leaving the Institute but I'm used to it now.
I'm a Brotherhood of Steel fan and I've never been with the Institute, so I have no idea what the Institute has as rewards, but I know the rewards from the BoS from Call to Arms to Molecular Level:
