With a sine wave they are all different. For the sine, if we take the peak as unity, the rms is 70.7% and the mean is 63% of the peak.

Things designed to run from a sine wave will be designed on these factors.

For an example if we think about a 100v nominal sine wave supply ( 100v rms) then the peak voltage will be about 140v. Many devices work from rectified dc so the dc of such things will be just under 140v if they have smoothing capacitors.

If you run a 100v lamp on a 100v square wave it will be fine as lamps work on rms and that will be 100v. your radio will now produce a ht voltage of under 100v instead of 140 and may not work at all and certainly will not work properly.

So called modified sine inverters have a square wave with about 50% duty cycle and the trick is to make the peak voltage about 140v. This is fine for the radio and will charge its capacitor to 14ov peak. It so happens that the rms value of this waveform is 100v rms so the lamp will also be fine.

The result of all this is that square waves will not suit much other than lamps and heaters. Modified sine will suit lamps and heaters and also many electronic things that rectify to dc.

Things with universal ( series) motors will work perfectly well on any of these waveforms ( and even dc in many cases)

There are other devices that fall into a very grey area and may or not work on modified sine ( even less likely on square wave).

Induction motors may or may not run, they will certainly run hot and rough on MSW and are worse on square wave. The harmonics cause circulating currents and also cause tendencies to crawl at sub-harmonic speeds. Laser printers and many small power supplies will be damaged by msw or square. Power tools with speed regulators may not work.

Circuits with electronic power factor control such as hf lighting ballasts may be totally confused and not work or blow up.

Most people seem to be lucky, but there is always an element of risk with most clever things with msw and the risk is very high with square wave.