By the majority of people, they both fall into the RPG category. JRPGs aren't western RPGs, but they both fall into the domain of role playing game.
I wasn't aware it had been firmly established that the "majority" believe one way or another. From my experience, in fact, quite the opposite is true: there are clear differences in design between cRPGs and JRPGs, just as there are certain elements which are similar. It's like saying first-person and third-person shooters are in the same category - depending on how broad your definition is they might be, but if we get right down to the specifics they really are quite different.
Let me start by saying that I really enjoy both styles of game (and indeed
all kinds of games) but ultimately I play JRPGs and cRPGs for vastly different reasons.
First of all, cRPGs are less about raw statistics and more about what those statistics represent, ie: the uniqueness of your character. There is a radical difference between a character with high intelligence and one with high strength, etc. Really great RPGs will even reflect this difference in the dialogue options that are available, or in how NPCs react to your character. In contrast, JRPGs generally only care about stats as far as they affect combat, and NPCs more or less react the same regardless of your character build. They do share some features like gaining levels and increasing the abilities of your character through experience and so forth, but their overall intent differs.
Second, JRPGs rarely offer branching storylines or variations to complete objectives, with only Chrono Trigger as the notable exception, and even then only in limited terms. A really deep cRPG will instead offer various routes to success, and the choices your character makes on their journey reflect their chosen personality and traits. This goes back to the pen-and-paper traditions of having a game master that could tailor a campaign to the input of his group of players, offering a great range of actions and resulting consequences - greater than is possible in even the most complex cRPGs that try to emulate this behaviour. This is completely unlike JRPGs, which are very rigid in their presentation of the storyline, where the player usually has no control over which direction events should take. If there's no choice in the outcome, how can the player have any real influence in playing their role?
A small sidenote: there is a great difference between being able to do a particular quest in various different ways, and having the option to do various different sidequests in only one possible way. Having a binary choice between doing a quest or not is no replacement for having real control over your character's modus operandi.
Third - more than anything the divide between JRPGs and "Western" RPGs is cultural. Like the difference between fantasy and sci-fi, there are certain conventions that define each uniquely: Japanese RPGs tend to have characters that rise to near-godlike status and solve all the problems of an entire world with one efficient quest, and while this is sometimes also true of cRPGs, there's generally more involving character development and depth to the setting. Where in a JRPG you usually set out to find a magic sword, mystic sage, and/or ancient prophecy in some form or another, cRPGs vary between such storylines as discovering why your brutish character cannot die, the intricacies of vampirism in modern LA, or the consequences of blindly following prescribed virtues.
Of course, though the general traits I've listed have been pretty well established, not every cRPG follows the same mold, and neither does every JRPG. The Witcher and Mass Effect have less flexible, largely-predefined character options, but the trade-off is that their storylines have a context which better reflects their central character (rather than the more generic responses in, say, Elder Scrolls games.) It's a fine balance between offering the player enough freedom to define their own character through their actions and unique abilities, and also having exciting, cinematic plotlines that motivate the player to keep following the story.
In the end, the real depth of an RPG comes down to how unique each character can be, and whether or not you can have a different experience each time you play. If there are only a few possible builds or outcomes, then it's more of an adventure game where the player follows a preset path and responds to stimulus as it arises, but ultimately cannot affect the final result. The hero in Dragon Quest 8 may choose spears over swords, or may have Angelo specialise in bows instead of staves, but it's not possible to avoid parts of the story or choose another path (except the Dragovian one for the alternate ending, but that's a different matter.) On the other hand, Fallout can be completed within fifteen minutes because the central "plot" isn't relevant to the 90% rest of the gameworld and the effect a player's character can have on it. You could say the difference is that in JRPGs your character is always at the center of the events that occur, and in cRPGs they are more of a random bystander in a world which exists despite them.
TL;DR - There
is a difference between "Western" cRPGs and JRPGs.

Go to almost any gaming website. They will likely have a "Role Playing" or "RPG" section. JRPGs and western RPGs are both lumped into that category even if they are notably different.
I base my observations on my experience with games from the last couple of decades, not what may or may not be written on a gaming website for sake of convenience.
