There's a huge difference in terms of input protection, actually. Fusing and a bunch of other protection circuits are the first line of protection... most cheap meters will have a glass fuse and little else (some not even that https://www.youtube.com/watch?v=glzLlPO6CjY), while a better meter will have MOVs, PTCs and many more lines of defense. Nicer meters (especially those rated for working on really high voltages that can cause a literal explosion) will have metal blast shielding around the inputs. There's also isolation- on cheap meters it's pretty common to see poorly laid out circuit boards that have input traces too close together. High voltages will arc right across and boom.

The nice meter I have would probably blow a fuse if you connected it to the wall on the current setting, but it's a fast fuse so the rest of the meter would be fine. The fuses in really crappy meters are often quite slow though, so they will blow and cut the circuit but not before the meter gets fried.

There are different levels of "crappy" though. A lot of the cheap meters you can get nowadays are safe enough to jam in the wall, I'm not saying "oh you've gotta spend $1000 on a multimeter or you're gonna die." But if you're gonna probe high voltage you'd better get something nice. I have a $5 meter that claims it'd be good at 1000 V, but that is obviously a lie and would be suicide.