Players that are the starter in a game get a kind of a multiplier bonus to stamina, allowing them to pitch longer than they would be able to if that came in as a reliever. On the other hand, relievers get a bonus to their Stuff rating that is determined by the strength of their best 2 pitches compared to their other pitches (representing the idea that they don't need to mix in their other weaker pitches because they will only see a batter 1 time usually, unlike a starter that goes through the lineup multiple times). You can see this effect on Stuff by changing a player's role in their action menu. Though I prefer to think of the reliever bonus stuff as a starting pitcher penalty. So if I have 2 pitchers capable of being a starter, I must consider who suffers the largest penalty playing as pitcher (it'll be the pitcher whose best 2 pitches are much better than the others), and also who I want to have a stamina bonus to be able to pitch more (I want the better pitcher throwing more)
Relievers will generally get better results against batters because of that stuff bonus and because of the fact that there's a bigger pool of relievers (anyone can be a reliever), but only a subset of pitchers can be a starter. Starters need more stamina (I think 25/100 is the minimum required by the game, I've heard?), and they generally need 3 or more pitches that are decent compared to each other. If they only have 2 strong pitches, but the 3rd is weak, they would get a large bonus to stuff if they are a reliever, and so would be giving up that large bonus if forced to be a starter.
The idea behind stopper is that you may want your best guys to come in when there is trouble, since that may be when their strong abilities will have the most effect on the outcome of the game. The idea behind closers is that you use the closer in the last inning when holding a lead because you know their success in the inning will be valuable because it gives you the win, whereas using a reliever successfully earlier in the game may or may not be important depending on if that scoreless inning made the difference in the game's outcome. If your team scored a bunch of insurance runs later, or if the other team took the lead later anyway, that earlier inning performance was wasted. So both stoppers and closers are different ways to try to maximize the value of an elite pitcher.
As for what ratings are important, it can depend on the situation. With a 1 run lead, a homerun would be big, so movement is quite important then.
If there are runners on base, especially runners on 3rd, and there are less than 2 outs, then stuff is very important because you want the strikeout so that the runner can't score on a groundout or sacrifice fly. So Stuff would be very important to a stopper, I think.
If there are runners on consecutive bases, especially bases loaded, then control becomes very important. Again, this may be more likely to be the case with a stopper.
If there are no runners on, then control may be less important, as you can maybe afford to give up some walks. This going to be the case for the closer that comes in at the 9th inning. So if you have great pitcher that just lacks control, he perhaps may be used most efficiently as a closer instead of having him try to get you out of a jam in an earlier inning where his lack of control might make him only marginally better than your other more balanced options. But this kind of closer might be okay with the 8th+ setting, where he'll come in late in the 8th if there's trouble, assuming he will be able to get through the 9th as well without getting tired.
If you have a more balanced superstar reliever, with high stuff and pretty good control, then I would put them as stopper.
Another factor is that you want your best relievers to pitch more often, just because they are better. But you need to balance that against needing your best pitchers to be rested and available for important situations. I mean, if your middle relievers are poor and giving away games early because you have too many setup/stopper/closers sitting in the bullpen well-rested because you assigned those roles to your best guys, then you probably would win more games having more of those late guys be assigned to the top middle-relief position, even if it might mean your late game superstars are unavailable some games due to being tired. Your weaker middle relief guys should still have a good chance to put the game away in those cases. That's better than having your stars go mostly unused.
Left/right matchups are nice too. If I have a lefty and righty that are roughly the same in terms of overall ability, I like to try to put them with the same setting (maybe both middle-relief and use-more-often, or both setup secondary, etc.) so that the AI will select the better L/R matchup when possible.
If you have a player that is really good versus only lefties or only righties, and you really don't want him pitching against others, you can use the specialist role. It'd be good to also use this player for long relief, since he can help eat up innings in blowouts, and he otherwise doesn't usually get that much use. Maybe he faces one batter and that's it, when put in as a specialist.
Stamina is nice on a long reliever to help minimize the need for anyone else to pitch during a blowout. Though stamina helps for any player so they can be used more often, but that can be a bit of a luxury to find on a good reliever.
I made another post on pitchers recently
here.