There have been a lot of writings on the difference between RPGs and Action RPGs, but I'll try to sum it up briefly. An action RPG is normally considered an action game with RPG elements, such as stats, quests that give items and experience, and level-ups. Damage per second becomes the most important factor in many action RPGs. Diablo is one example of this as much of what you do is explore a dungeon and click furiously whenever you see an enemy. They are fast-paced games where reaction speed becomes important.
However, RPGs differ from Action RPGs in their slower pace where tactical thought and resource consideration is more important than reaction time. Mental challenges such as riddles and puzzles are frequently used to allow clever players to solve quests without having to rely on brute force. Many RPGs are turn-based because of this focus on beating difficult encounters through intelligent play.
Action RPGs are such games as Diablo, Nox and the World of Warcraft MMO.
RPGs are such games as SSI's Pool of Radiance, Might and Magic, Planescape: Torment, and Fallout 1.
Ironically, Diablo was based on rogue-like games, but those games are slower paced and button mashing will often get you killed, unlike in Diablo where button mashing was often necessary. Also, rogue-like games such as Nethack have their own internal puzzles that Diablo lacks.
Games such as Oblivion, Fallout 3, and Skyrim walk the line between an action RPG and regular RPG. Often these Bethesda games are simple button mashers, but they do occasionally have quests that can be solved without fighting or being ridiculous fetch quests.
Action RPGs and action games are being blended so often that they have more in common than action RPGs do with normal RPGs. Mass Effect (2007) has more in common with Grand Theft Auto: San Andreas (2004), than it does with Temple of Elemental Evil (2003).
Here's TV Trope's take:
http://tvtropes.org/pmwiki/pmwiki.php/Main/ActionRPG
"Action RPG
A
Role-Playing Game mixed with
Action Adventure. While these games retain many
RPG Elements such as experience, levels, statistics, giant inventories, and sometimes even random battles, the gameplay is primarily that of action. You run up to an enemy and hit it with your sword, rather than selecting the "Attack" action from a menu. Often this means seamless interaction between you and the enemies, rather than thinking of two different parts of gameplay (the exploration and the battles). Think
Zelda but with more statistics, experience, depth, and people to talk to. Battle menus can still exist, but generally need to be manually brought up rather than an integral part of the battle process. This genre is essentially a subset of
Eastern RPG; it attained massive popularity in Japan in the mid-1980s. While many
Western RPGs have mixtures of battle and action, this is much more built into the definition of a
Western RPG, and hence less of a separate genre. Eastern games, on the other hand, tend to be more firmly in either the "traditional RPG" or "action RPG" camp. There are some games that blur the line between the two, though, featuring a separate battle system which nevertheless is action-based (so you may wander the world and get into a random battle, in which case you suddenly have to be on your toes). Compare
Beat Em Ups, in which Action
RPGs derive their basic gameplay, but do not contain the
RPG Elements."
