They're not spikes. Through a purely resistive load, the current is at zero/max/min whenever the voltage is at zero/max/min. But inductive loads make it so that the current lags the voltage (the current reaches zero/max/min after the voltage does).
And you can see that this means that the power isn't the same anymore. For example, in a resistive load when current and voltage peak at the same time, that's when there is the most power since V(t=T)*I(t=T) is largest since max V and max I occur at the same time and are multiplied together, but if the current or voltage is shifted a little bit, the V*I is smaller, since now the when the maximum of the current and voltage no longer occur at the same time, so the highest V*I that you can achieve will be lower.
Imagine two identical sychronized sine waves graphed out, and multiply them together to make a 3rd graph. Now imagine that you shifted one of the first two sine waves a bit. Now multiple the shited and unshifted graph together. The graph made from two synchronized sine waves (kind of representative of power) will have higher peaks and different values than the graph made from unsynched waves.