In the "you are welcome" form, you can use it as you would use "Oh don't mention it". It comes from the other usage, which is used to express that something is not realistic/desired: "Ma figurati se adesso mi metto a insegnarti l'italiano gratis", I guess it can be conveyed in English as "But imagine if I'm going to teach you Italian (how ridiculous/inconvenient/annoying that would be)", although in the English world one would use sarcasm instead. In the welcome form, it becomes "oh imagine (if you needed to seriously thank me for this, how superfluous that would be)".

Figurati is used when you adress yourself to a person in the second singular form : TU for person you know and are befriended with Si figuri is used for the LEI. formal way of addressing somebody with respect.

The difference is very simple: it's just the formality. With both expressions you are talking with one person only. With "figurati" you are saying "you are welcome" to a person in an informal way, so a friend of you or a person that you know and it's not necessary to be formal. "Si figuri" it's the same but the formal way. You are talking with someone (just one person) and you want to be formal. I hope it's clear now the difference. Two expressions, same meaning, one informal and the other one formal. :)