Speakers do not clip and speakers do not 'need' power.
Speakers act as a reactive load (resistance, but varies by frequency.) Your amplifier takes an input signal of a given voltage amplitude and multiplies it by whatever gain it is set to, and then can either maintain that amplified voltage on the outputs or it can't. It can clip if that gain-multiplied voltage exceeds its power supply voltage limits - it 'hits the rails' - and this can show up even without output load, as Amir frequently complains about in AVR testing. Or it can clip if attempting to apply that voltage to the speakers' reactive load exceeds its power supply current delivery capacity.
When the amp clips, to simplify things, the normally sinusoidal output waveform distorts at the peaks. This produces distortion of varying order. It does not produce 'lumpy frequency response'; if an amp's FR is bad it is generally a function of the negative feedback circuit being messed up (or interacting with the load when it shouldn't, which isn't necessarily volume dependent.) That clipped signal also acts more like a DC output which ends up causing more heat generation in the speaker voice coils, which can contribute to 'frying' them.
You can talk about speakers with 'difficult' highly reactive loads but it's not really needed for this discussion.
Speakers can suffer from power compression, i.e. running them near but not necessarily at their limits can also cause voice coil heating leading to the motors become less efficient and the output volume to drop (and in the worse case causing the user to crank the volume to compensate, making it worse.) Also if you're really pushing your drivers you can get into non-linear ranges where it's not quite hitting mechanical limits but the physical resistance of the speaker is making it harder to move further - don't do this, and certainly don't go beyond it.
Basically like most machinery you want to size your amps and speakers such that you're comfortably within limits during all operating periods.
@Spkrdctr is correct above but the opposite side of the audiophoolery is that, while you may not need a huge amp to supply a tiny load to your speakers, it is also not necessarily true that a smaller wattage amp is better at it (i.e. the Nelson Pass First Watt business.)