Distinction between "ambulate" and "perambulate"?

At a cursory glance, these two terms appear to be synonyms. But that would strike me as unusual since one is a prefixed form of the other. (Prefixes are used to modify a word in a specific and consistent way. As a result, the prefixed form of a word has a different meaning than the word.)

"Per-" confers a quality of thoroughness or fullness. But in the context of ambulation, this doesn't seem like a significant modifier (to me).

Am I wrong, or are "ambulate" and "perambulate" synonyms?

I'd say you are wrong.

To ambulate is to move by means of walking, parallel to the way brachiate means to move by means of swinging with your arms from hold to hold.

To perambulate is to walk through, or to traverse, something. Perambulate therefore can take an object to be traversed. You can "ambulate idly all afternoon", but rather than "perambulating idly all afternoon"; you should "perambulate the yard idly", or some such thing.

I cannot show any evidence for this but I suspect that perambulate does not use the prefix per- meaning through, but instead a shortening of the prefix peri- meaning about (Greek περί but also used in Latin).

Every case I know of perambulate suggests returning to the starting point, either wandering around (i.e. without a destination) for the purpose of exercise and fresh air (hence pram) or surveying boundaries by walking them.