Oh man, this is a big topic. I'll tell you a few of the things I've heard as I've tried various cartridges.
None of them sound exactly alike, they all have some kind of sonic character. I know, this is evidence of an imperfect cartridge, as ideally they should all sound the same and perfectly neutral. But we don't live in an ideal world, unless we happen to have about $15,000 for a top-of-the-line Koetsu, then things might become ideal.
One of the first things I noticed when I upgraded to a Dynavector 10X5 was much better resolution, it sounded as if someone had opened up a window, ie much, much clearer. At the same time you'll probably notice better frequency extension, depending on what you're using, but IME at first it's most noticeable in the higher frequencies.
Then of course there's detail, more of it, the noticeable kind you already knew was there but can now hear better, and possibly the inner detail in a song you've been listening to for years, yet you never noticed it before. Imaging improves, meaning you'll hear better separation between instruments, and hopefully more "air" or space around them. I found this to be the most striking improvement with my Grado Platinum, and I love it. This can go hand in hand with soundstage, where things sound like they're placed in specific spots, a more three-dimensional sound. Some cartridges struggle with this, and a lot of it also has to do with speaker placement and room acoustics.
The better cartridges I've heard have better dynamics, but this is not necessarily simply about impact. I don't know about you, but a lot of times when I think about dynamics in relation to audio I immediately start thinking about the impact of, say, percussion instruments. But there's more, or there should be at least. Cartridges described as more "refined" (often used to describe MC cartridges) move away from the loudness of bass, for example, trading it for definition and realism. Of course MM cartridges can do this too, but the ones I've heard tend to do the impact aspect of dynamics better than definition and realism. This is NOT to say there aren't lots of MM cartridges that provide excellent definition and realism.
Although not always the case, what you should have with a better cartridge is a better tracker. I can't think of one instance where my Dynavector mis-tracked, even on very warped records. On the other hand, my Grado Platinum, as much as I love it, is rather disappointing in that regard.
Finally, and not insignificantly, it should look cool. :thmbsp: