I've been a High School baseball and softball umpire for years, so here's some background on those pitches.
The spitball was banned in 1920 after it killed a batter. By putting petroleum jelly or spit on a ball, it makes it lopsided. This makes the ball wobble a bit when it is pitched. It makes it harder to figure out where it's going to end up. That, and the fact that they re-used the same ball for the whole game, made it really hard to see the pitch. The batter who was killed did not even react to the pitch because it was so hard to see. The MLB finally banned the spitball, and now a ball must be replaced when there is any sort of mark on it. And I'm completely fine with that rule.
The screwball is not illegal, it's just uncommon because pitching it too much can really mess your arm up.