[robocat]

As a New Zealander, I wouldn’t say either of those words were similar, although I admit our usage of English is special.

My naive interpretation of those words is: perambulate is closer to just walking; rambling is a country walk with the sense of avoiding normal walking routes, but often planned and often walking with others (really need UK person to clarify).

Wander or meander-through have the sense of randomness, but lack the intentionality of deep attention to your surroundings.

[notahacker]

Brit here: can confirm that rambling generally implies countryside walking (perhaps a bit slower and less purposefully than hiking or trekking). Perambulation sounds a bit Victorian and tends to imply walking in some sort of defined circuit. Strolling might be closer but also doesn't really imply paying much attention to surroundings.