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.
- A New Dawn is the BoS's epilogue quest, after you defeat the Institute. You get a T-60 Jetpack, which ain't too shabby.
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:
- Building a Better Crop gives you 20 stimpacks. Whoopty-doo.
- Plugging a Leak gives you Liam's Glasses, which is the best thing the Institute will ever give you. They give +2 to Int. Unfortunately, you can't get finish this quest if you've done Tradecraft for the Railroad, which gives you the much better Deliverer pistol. Tradecraft is also needed before you can do the Railroad quests that allow you to build ballistic weave.
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 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.