The dictionary definitions of these three words aren't particularly helpful in figuring out when to use which one.
Lewis Elementary's definition of sānē includes
- indeed, doubtless, by all means, truly, certainly, of course, right, very
Certē, meanwhile, includes
- really, surely, assuredly, actually, certainly, as a fact
Finally, profectō is
actually, indeed, really, truly, assuredly, certainly.
This is very frustrating from a living-Latin perspective. Obviously, we use "certainly" in a lot of different contexts, but I wish the dictionary had more to say about which contexts call for which word. From what I can tell, for example, one use of sānē is as a concessive "to be sure"—"He's loathsome, to be sure, but he's still your brother." But I can very easily imagine that "sure" in "Surely you can see that I'm right about this" would call for a different word.
The example sentences provided in Lewis & Short aren't much help either. In Nōn est ita, jūdicēs, non est profectō, profectō could be meant in so many different ways that I find it unhelpful as a guide. ("It's not like that, judges; [you may think it's like that, but] in fact it's not." Or "It's not like that, judges; it most assuredly isn't." And so on.)
Can anybody provide any guidance, however incomplete, in the matter of different uses of these words? It's possible that the clearest thing to do would be to provide English examples illustrating the different uses (i.e., "You would use certē if you were trying to say something like, 'Blah blah blah certainly blah blah,'") but I'm happy for anything that helps.