Sine wave inverters are always more expensive - from 2 to 3 times as much. A modified sine wave inverter actually has a waveform more like a square wave, but with an extra step. A modified sine wave inverter will work fine with most equipment, although the efficiency or power will be reduced with some. Motors, such as refrigerator motor, pumps, fans etc will use more power from the inverter due to lower efficiency. Some fluorescent lights will not operate quite as bright, and some may buzz or make annoying humming noises.
Many appliances get their timing from the line power.
Because the modified sine wave is noisier and rougher than a pure sine wave, clocks and timers may run faster or not work at all. Items such as bread makers and light dimmers may not work at all - in many cases appliances that use electronic temperature controls will not control. Very few but the very cheapest inverters any more are square wave. A square wave inverter will run simple things like tools with universal motors with no problem - but not much else. For an inverter to run, it must have batteries.