These words are the same when used this way, yes. Continuing can also be used as a verb, but continual cannot. Continual is more often used for something which repeats or has a long history of continuing. Continuing is more often used for something which is constant or has only recently begun.

they mean the same but, continual rain = rain which does not stop, emphasis on the rain itself continuing rain = rain which continues (from a certain point onwards), emphasis on the time when the rain occurs I think 'continual rain' is probably the more common way to say it, as we usually describe the rain's quality, not when the rain occurs, (at least when we use a single adjective to describe it)

I agree that the distinction between the two depends on context. I think the usage of one term over the other would really depend on how it sounds, and not the actual, technical difference in meaning between the two. For instance, you could even say "...continuous rain..." and not change the conversational meaning. Something else you might consider, again, depending upon the context in which you're using this term, is saying "...constant rain...". To me, "...constant rain..." sounds a little more natural in a conversational context.

I can't see a distinction here.

I think they are of the same meaning. And of course it depends more on the context it is ued.