They vary a lot. The below is by no means an exhaustive list; it contains just the highlights of what I've found from perusing the INI files and a bit of the code.

Aim cheating

^{Found in: XComGameCore.ini, AimAssistDifficulties}

On every difficulty except Legend, the game applies a number of modifiers to both your attacks and the aliens' attacks. There are five categories:

• Base aim multiplier (Rookie and Veteran): Your shots have an invisible multiplier applied to make it easier to hit.
• Miss streaks (Rookie, Veteran, and Commander): For each consecutive miss in a turn, your soldiers are given an invisible aim bonus.
• Hit streaks (Rookie only): The aliens take a penalty to aim for each consecutive hit they have in a turn.
• XCOM hit chance for dead soldiers (Rookie and Veteran): For each soldier you've had die on the current mission (below 4 soldiers), your soldiers are given an invisible aim bonus.
• Alien hit chance for dead soldiers (Rookie and Veteran): For each soldier you've had die on the current mission (below 4 soldiers), the aliens take a penalty to aim.

I'm not sure why Commander gives you a miss streak bonus (which is actually higher than Rookie or Veteran). Judging by the comments in the code, it's an oversight and might get patched out.

Other tactical game tweaks

• Enemy stats, mostly health and armor but occasionally also crit chance, dodge, and aim (maybe more)
• How likely your soldiers are to be shaken after a mission, and how many missions it takes to recover from being shaken
• Whether the alien AI goes easy on you; aliens will limit their number of attacks per turn on easier difficulties
• How long it takes your soldiers to heal from injury (significantly longer on Legend)

Strategic game

^{Found in: XComGameData.ini, all over the place}

Almost too much to list here.

• Rookie recruiting costs
• Research time, soldier recovery time, and the time to clean rooms on the Avenger
• How quickly the Avatar Project makes progress, especially based on how many facilities the aliens have
• How quickly the aliens adjust their "force level", which influences what enemy types can spawn on missions and what kind of loot you can find
• Whether you get bonus mission reward intel at regular events (Rookie and Veteran only)
• How much rushing research at the Black Market helps you (halved on Legend)
• The chance you'll get given a reward you "need" (e.g. a Scientist or Engineer if you're low on them, or soldiers of a particular class, etc)
• The intel costs of items at the Black Market
• The initial game difficulty and how much difficulty increases month-by-month
• The relative likelihood of which Dark Event you will receive, and how much intel it costs to reveal a hidden Dark Event
• How long it takes to train a rookie, respec a soldier, or complete psi training
• How many corpses you need to complete an autopsy instantly
• How many supplies you get per region per month, and how long it takes to make contact with a region or build a comm tower

Many of the changes are designed so that Legend, which starts out harder, doesn't quickly scale into an unwinnable mess. For example, general difficulty starts higher on Legend, but it grows more slowly than even Veteran does. Legend will still remain ahead of Veteran even with the scaling, though.

The Laboratory (you can only build one) functions in a same way as the Workshop: It makes a scientist count as two scientists when he or she is assigned in the Laboratory (you can assign two after upgrading), just like Workshop creates two drones per assigned engineer.

Having no more than two scientists but both of them being assigned in the lab will decrease research times for the same amount as having four scientists but no lab.

There's also a continent bonus that decreases research times just by having the laboratory built. It gives 20% faster research rate when it is constructed, giving it some additional boost.

In terms of how important building the Laboratory is, well I'd say with difficulties other than legend not very. The bottleneck is more often lack of supplies/alloys than available tech. With legend difficulty however research takes so much longer that you will have much more supply drops before you get the next level gear unlocked. The available tech will be a more significant bottleneck than lack of materials. Therefore having either a lot of scientists or the lab if you did not get many is a necessity. For example, magnetic weapons research with 0 scientists takes over 60 days in legend, over 40 with one scientist and still over 30 with two scientists.