It's a diminutive form, you can say. We also use it in Portuguese with several words.
I don't know exactly how to explain why we do it, but we use it a lot. So, I guess, Spanish speakers do it too.
Sometimes it's to make the sentence sound more softly, delicately, gently... Something like that.
Diminutives were also used a lot in Latin, too, so it makes sense that the Romance languages would make good use of them.
An example in Latin is the word 'avunculus'. It means maternal uncle (French 'oncle', and by extension English 'uncle,' are descended from it), but it comes from the word 'avus' meaning grandfather. So it literally means little grandfather
Nevera vs. refrigerador vs. frigorífico vs. refrigeradora vs. heladera? :)
Despacio vs lentamente isn't really a regional thing, but there are some differences in meaning between those two. While lentamente only means "in a slow manner", despacio also has the meaning of "bit by bit", "carefully", or "quietly". Lentamente is used more to describe processes, while despacio is more frequently applied to people.
Actually, it is grammatically correct, since slowly us an 'ly' adverb. But it is not uncommon to hear "Slower, please" also.