After each competitive game you gain or lose skill points. How is this number calculated? I've heard payload games and defend-the-point games have different values. Is this true? Also, is it just win/loss that goes into the calculation, or do other things like a close game or medals won factor into it?
Its not a flat + or -, loads of factors come in play.
Did you win or lose does have an effect, but there's also your calculated team contribution that changes things up.
Overwatch, like other comptetitive games, uses systems that are related to the ELO ranking. https://en.wikipedia.org/wiki/Elo_rating_system
Basically, it takes into consideration your level, the level of your opponents, and then (the process is pretty complex) if you beat a higher rated opponent, you win a lot of ELO, if you win against a lower rated opponent you win elo, but not as much, etc.
The amount of SR gained/lost in competitive matches is directly related to the MMR (matchmaking rating) level of your team vs. the MMR level of the enemy team. MMR is a hidden stat tied to each individual player, and it is not viewable. Basically, it's a metric Blizzard uses to determine how skilled (in general) you are at the game.
If your MMR is lower than the enemy and you win, you will gain an increased amount of SR.
If your MMR is lower than the enemy and you lose, you will lose a decreased amount of SR.
if your MMR is higher than the enemy and you win, you will gain a decreased amount of SR.
If your MMR is higher than the enemy and you lose, you will lose an increased amount of SR.
If your MMR is approximately equal to the enemy and you win/lose, you will gain/lose the normal amount of SR.
In the situations above, "increased", "decreased", and "normal" are general terms that describe the relative amount of SR you'll end up with, but I don't know the exact numbers.
Blizzard has not (and probably will not) ever released the exact formula used to determine how much SR is gained/lost, so this is as close an approximation as we can figure.