You can use either of those sentences to say that you have arranged to visit your mother tomorrow. They both clearly express that meaning.
I'll be visiting my mother tomorrow
somewhat emphasises that tomorrow you will be busy (visiting your mother), so you won't be available to do other things. Doing this thing will be in progress, so other things can't be in progress.
So, you might slightly prefer to use this sentence when someone invites you to do something tomorrow, or requests that you do something tomorrow, and you want to politely explain that you have arranged to do something else (you will be doing something else).
It might be a bit clearer, because we tend to use future continuous forms when we want to specify the time when we expect to be doing something. Consider:
"Are you coming to the meetup tomorrow (at four o'clock)?"
"I can't, I'll be visiting my mother (at that time)."
But, especially if this is an informal spoken conversation, you could still politely refuse using the present continuous form; there won't be anything incorrect or rude or ambiguous about it.