One way to think of clipping is that it is limiting with a compression ratio of infinity:1 and a threshold of 0 dB (or +12dB, +18dB, depending). Thus, signals that rise over that threshold are "clipped", completely prevented from rising over that threshold.
When limiting, typically a very high compression ratio is used, but infinity:1 is uncommon. Also, the threshold is typically set a little lower the maximum level the hardware can handle. This allows a little bit of headroom, so that when a signal rises above the threshold, it can still rise, but it is made to rise slower.
However, the primary difference between clipping and limiting is that limiting is done with by looking at the average of the signal. That is, a limiter does not consider each peak (electrical peak) of a signal, but rather takes a small average. Additionally, a limiter has response times on both its attack and release. These give the signal more freedom, resulting in a still musical sound.
Clipping, on the other hand, is something that occurs at the immediate amplitude of the signal. In other words, instead of looking at an average and dealing with attack and decay, a piece of equipment that is clipping simply cannot deal with such a high signal level and therefore at each electrical peak of the signal, it maxes out. It simply cannot produce a voltage high enough. In a sense, clipping is at "less refined" level, if you will. Each individual peak is clipped without consideration of other parts of the signal, whereas a limiter considers a larger part of the time domain.
Hope this helps, I just realized how difficult this is too explain.