First and foremost, very few words in English are "perfectly" interchangeable.
still (adv.) up to and including the present or the time mentioned
yet (adv.) up until the present or an unspecified or implied time
I hadn't thought much about this before, but using the word yet suggests a glimpse into the future:
I can't speak English yet – but I won't quit trying until I do.
while using the word still suggests a glimpse the past:
I still can't speak English – even though I've been trying for 10 years!
I'll try this again; the quotes here are in italics, what follows in [brackets] is what I might infer from the speaker's choice of words:
The bus hasn't come yet [but I expect it will come soon].
The bus still hasn't come [I've been waiting such a long time!]
I think you can even combine both words to express exasperation:
We've been potty training Dora for six months now, but she still hasn't got it yet!
That wording indicates it's been a long time, but there's still hope the desired result will happen eventually. Similarly, going back to your original examples, one could say:
I still can't speak English yet!
By the way, this answer hasn't even mentioned the use of these words to mean "even", as in:
We'll have even more snow tomorrow.
We'll have yet more snow tomorrow.
We'll have still more snow tomorrow.
That's another context entirely.