If you go by Wikipedia,
In topography, a summit is a point on a surface that is higher in elevation than all points immediately adjacent to it. Mathematically, a summit is a local maximum in elevation. The topographic terms "acme", "apex", "peak", and "zenith" are synonyms.
According to this glossary,
The summit is the highest point.
In theory, every mountain has exactly one summit. In practice, mountaineers will in many cases talk about multiple summits (eg Everest South Summit), thus blurring the distinction between "summit" and "peak".
Sometimes as you're climbing you'll approach a spot that looks like it might be the summit, only to discover it's a "false summit".
The word comes from Latin "summus", meaning "highest".
In the 19th century, "summit" was also applied to other kinds of highest points, such as the highest point reached when crossing a pass.
The same glossary has an entry for peak:
A peak is a point that's higher than all other adjacent points. In mathematical terms, it's a local maximum, the point with slope of zero along a convex-up curve. The thing to understand about this definition is that it is entirely localized: there may be some higher point not far away, but if you can't get there without going downhill first, you're standing on a peak.
In other words, most mountains will have multiple peaks.
So, to sum it all up, unless you are particular about not piquing the interest of some topographical fusspot, it appears that there's nothing wrong in using the two terms interchangeably. However, technically speaking, a mountain can have only one summit, but multiple peaks. It follows that the lone summit is also a peak :)