What does StarCraft 2 campaign difficulty affect?

As you go up with difficulty:

• Game speed increases. Brutal is on "Fastest" and each difficulty below that is likewise one speed setting lower.
• Computer AI gets better. Brutal opponents will target your medics first, if possible, for instance, whereas normal and even hard AI opponents will be content to fire at the marines being healed by them.
• AI gets more preplaced units, and attacks with larger groups of units. Example: 2nd Mission: the mineral / gas cache above your start location is unguarded on normal and casual, but on Hard, you've got a marauder and two marines. On brutal, you've got hammer securities and war pigs instead (the stronger, merc versions, of the respective units)
• Some achievements become attainable. No mission achievements are attainable on casual. Normal nets you an achievement for completely finishing (inc. bonus objectives) each mission, as well as a specific achievement related to the mission itself. Hard offers a third achievement, and brutal difficulty is required for the "beat all missions in the campaign on brutal difficulty. (duh! :P)
• Enemies start with upgrades / upgrade sooner
• Numerical objectives may also increase: the number of opponents you must survive on the "In Utter Darkness" mission increases with difficulty (and note that the achievements for this mission are based on kills beyond the mission requirements. The 3rd achievement for "In Utter Darkness" is much easier to get on normal, as when you earn the achievement on normal difficulty, you'll have just slain enough enemies to win on brutal difficulty)
• No change in cutscenes, etc, that I know of.

So, your interface is always the same. Marines always cost 50 minerals, and they behave the same way.

In harder difficulties, the AI is more agressive (attacks sooner and more frequently) and is smarter (i.e. better control over units), the enemies have more HP / attack, there are more enemies to encounter (Example, in the second mission, there's a little alcove north of the starting base with minerals and vespene pallets, on easier difficulties, there isn't any force there, on harder difficulties, there's a lot more), and harder enemies to encounter (on the third mission, on casual, my wife encountered zero mutalisks)

I don't think they give more starting money to computers, but I don't think they need to. On easier difficulties, the computer doesn't even use a fraction of the money it gathers.