Personally, I think it is all the same. Roleplaying without the usual limitations on things of a extremely dark and/or sexual nature. Perhaps it is more broad than that, perhaps not. However, people often see these terms differently. Some people think adults are better roleplayers than younger people (usually true, admittedly). Some people just want to have the option. Some people want it to be focused almost entirely on the sexual. Some people just want there to be absolutely no limits. Ultimately, all of them touch on sexuality to a degree (if you use a broader form of it to include virtually any fetish that brings about arousal) and that is hard to find in most venues of RP. Even on the Net.

I've primarily (as in, more often than not) seen the following terms used in the following manner:

Erotic Roleplaying

refers to roleplaying that involves strongly sexual themes. At the very least, the intention of engaging in such themes. Not all Erotica leads to sex. Not all RPs do either. In other words, Erotic Role Play, involves role play in a strong interest in sexual themes which may, or may not, materialise during the course of the roleplay in question. HouseEros (my site) caters to this type of RP but is focused on an 'all' situation.

Mature Roleplaying

usually refers to maturity in the sense of capable of handling mature themes. This could be anything from sex to violence to a conversation filled with subtext. The goal of the players involved being to play with people who act like adults and lack the embarrassment adults feel when they talk about sex to someone who isn't an adult. Some also are just looking to raise the quality of writing they deal with in their RP. HouseEros is not focused on this type either.

Sexual Roleplaying

is...complicated. This is primarily because it is often used to cover real world activities involving a couple acting out their various sexual fantasies in person. Some people will undoubtedly find HouseEros thinking that is what it is all about. Unfortunately for them, they would be incorrect. Literotica, HouseEros, (and similar sites) uses a different definition. The definition of sexual roleplaying, in our context, is that it involves explicit sex as a major element.

Adult Roleplaying

is by far the most common term. This, honestly, is what HouseEros is focused on. Roleplaying with other adults without the limitations of a rating. It can be all of the above, or none of the above. Adult Role Playing is essentially rolpeplaying without the usual limits one finds on other sites in terms of content and themes.

If you found this blog while looking for one of these terms, I would suggest visiting my roleplaying forum at House Eros, an Adult Sexual RP Forum