Great question, but there’s a problem: I don’t like either very much!
With that out of the way, I’d probably go for a good moving magnet cartridge over a cheap moving coil.
If you have say around $500 to spend, and assuming you are chasing a medium compliance cartridge for a typical medium mass tonearm, there are lots of choices. Moving coil cartridges are technically better for a number of reasons, but at this low price point (yes, $500 is low priced for cartridges), moving coils are almost always high output types, of basic construction and generally don’t sound great.
Why? There are no surprises here. Cheap cartridges with aluminium cantilevers and spherical or elliptical styli don’t sound amazing, whatever the flavour. That’s because they are cheap. A nice moving magnet at this price probably has a better diamond, better cantilever and is going to sound warmer, punchier and probably a little easier on the ear, even if it does lack a little resolution.
Much beyond $500, moving magnet cartridges don’t get a lot better because they are limited by the moving mass or inertia of the moving elements. Low output moving coils start to come into their own at around the $750 mark. From here, the advantages of moving coils become critically important and this is why most cartridges I recommend from $750 are MC types and basically all high-end cartridges are coils.
It’s also why all of my serious carts are and have been moving coils. There’s no accident or conspiracy here, you can just listen on good systems and you’ll hear it, but it creates that hi-fi equipment problem where one wants to spend more money!
Keep in mind though that you need an altogether better playback chain to get the most out of a low output MC cart. A real rookie error involves listening to a nice MC cart on a cheap phono preamp. You’ll never hear all that the low output MC cart can do because of the limited resolution of the phono preamp. The best solution is a step-up transformer.