The two main issues are efficiency and undesired interference from the additional harmonics present in a modified sine wave. That means that a pure sine wave inverter is good at two things: efficiently powering devices that use the alternating current input without rectifying it first and powering devices like radios that can suffer from interference.
Some useful questions to ask yourself to determine if you need a pure sine wave inverter include:
If you answered yes to either of the first two questions, you may need a pure sine wave inverter. If you answered yes to either of the second questions, then you may be fine without one.