While not apples and oranges, I've always thought Mozart and Beethoven were different enough to not really be compared.
As far as I can tell from my listening, Beethoven is fierce, passionate, and fiery. When you play Beethoven, odds are you slam the hell out of the keyboard.
As far as I can tell from my admittedly rather limited exposure, Mozart is a little lighter. Not in terms of emotional content. I sort of understand it in colors, if that makes any sense. Beethoven is deep, smoky red while Mozart is more of a sky blue. Mozart seems to me to be a little more playful, and a little less...bipolar, shall we say? Again, not the greatest terminology. When you play Mozart, odds are your fingers are light and quick.
Beethoven knocks around in steel-toed boots. Mozart skips around barefoot.
While I like Mozart, it seems that in many cases what we have is essentially 250 year old elevator music. He certainly wrote pieces that touch the emotions, I'm not denying that, but overall I would classify him as one of the more moving composers.
It's probably just my preferences, but I've found that Beethoven more readily reaches me. When I need to concentrate, I reach for my Mozart. He gives me something aesthetically pleasing to focus on so I don't end up listening to everything else that's going on.
When I'm feeling something intensely, Beethoven understands. For me, his music speaks in a way that I can't. Also, when I'm listening to him, I have to move. I'm a pretty kinetic person anyway -diagnosed with ADHD- but something about Beethoven, whatever it is, won't let me not move. I have to experience him completely, with all of me. Whenever I get some time alone -I share a room so it's a rare treat- I lock my door, crank up my speakers and leap around to some mastery.
I've never had a similarly intense experience with Mozart. I actually get kind of bored with Mozart, while with Beethoven I can listen to the same piece over and over again, and still be affected the same way.
However, to be honest, I will admit that my exposure isn't that wide. A lot of it's also been from the radio so I don't remember the names of the pieces, just the composers.