Miter saws are designed to make cross cuts in narrow pieces of wood. The maximum width that can be cut is on the order of 8-12", depending on the saw. Sliding compound miter saws (SCMS) have a slightly higher cut capacity but typically cost 2X more than a regular compound miter saw (CMS). Miter saws are great for cutting molding, dimensional lumber, floor boards, etc. Can't imagine when you'd use one on a MAME project, however.
Table saws are designed to make rip cuts in large pieces of lumber. You can use the miter gauge to make cross cuts, but the table saw really excels at rip cuts. It is also possible to make angle and bevel cuts with a table saw. Low-end table saws cost as much as high-end SCMSs. A table saw is a very dangerous tool if you don't know what you're doing. The fact that you don't know whether to buy a table saw or miter saw makes me think that you don't. Do not buy a table saw until you understand and know how to prevent kickback accidents! A table saw is a more general purpose tool than a miter saw. If you buy anything other than a benchtop table saw, you'll end up doing a major portion of the assembly, which means that the saw won't work perfectly right out of the box. If you don't have the tools and expertise to adjust a table saw, you'll need to find someone who can help you put it together. Finally, keep in mind that using a table saw to cut large pieces of wood (e.g. an entire sheet of MDF or plywood) is very difficult to do safely! The best strategy is to cut large pieces of lumber to approximate size using a circular saw, then use the table saw and a helper to cut pieces to their final size.
If you could pick only one of these two tools to own, a full-size (e.g. contractor's, hybrid, or cabinet) table saw would be the tool to own. Benchtop table saws have their place, but they really aren't ideal for building something as large as a MAME cabinet. If all you're just going to build control panel boxes, say, they might be a better fit.
And it may be that you just need to work on your technique. I've built some pretty decent furniture, kitchen cabinets, etc. and I don't own a table saw... A circular saw (albeit a Festool), router, a decent block plane, and some good measuring tools will go pretty far if you know how to use them!