If the cut scene is being rendered in the game then it would use the same
engine.  If the cut scene is a prerendered movie like in a Final Fantasy
game then it's normally rendered using a completely different engine.
It's rendered using a high quality professional raytracer (not normally
written in house, but a company like Square/Enix might be an exception)
at the rate of something like one frame an hour on a expensive "render
farm" of dozens of computers.  Normally different artwork, much more
detailed models and textures, are used in these raytraced cut scenes.

As far as prerendered raytraced cut scenes go, you don't see them as
much on the GameCube because there isn't as much room on the GameCube
discs to hold the prerendered movies.  I don't see why you wouldn't see
them in Xbox games, except for perhaps because the Xbox doesn't see a
lot of the games where you would have a lot of cut scense (eg. RPGs).

                        Ross Ridge

I am not saying that they use seperate engines, what I meant to say is
that they seem to be able to do more with the same engine when the
hardware is only being used to render a scene.  For example, when you
benchmark a PC game with pregenerated timedemos, the benchmark will
typically give you higher framerates than what you would get when you
play the game because there is no AI involved and the engine already
knows what it is going to render for the duration of the timedemo.

In Ratchet&Clank, I think that a video clip is being played in many of
the cutscenes even though it looks like it is being rendered in real
time.  The developer just tricks people into thinking that it is being
rendered in real time because the models are the same as the ones
being used in the game.  I read in many reviews where people were
amazed that the cutscenes were given credit for being rendered in real
time.  If you look at the drastic drop in quality between those
cutscenes and the actual game graphics, there's just no way that the
PS2 was rendering those cutscenes in real time.  The same engine might
have been used to render the cutscenes, but it's almost certain that
there was more powerful hardware being used.
In FFX, I notice what what you're talking about but the cutscenes
still seem to be generated in real time, as they are not several times
better than the in game graphics like some of the Ratchet&Clank
cutscenes.  Now that you mention it, I do remember that Xenosaga did
seem to look slightly sharper in the cutscenes but I don't know if
it's due to the closer cinematic angles or that they used better
textures and filtering in the cutscenes.

The PS2 tends to have more games that use cutscenes because it has
more japanese developers who like to use cutscenes and Sony has always
considered cutscenes to be very important.  One can argue that
cutscene quality was a main reason why the PS1 beat the N64.  The
Gamecube and the Xbox games seem to rely less on cutscenes, because
they don't have as many 3rd party japanese developers who like to use
cutscenes and the Gamecube media doesn't allow much high compression
video to be stored.