It does change things because in this case "speak" is a verb and "speaking" is a noun.
"You've frequently heard me speak of her." vs " You've frequently heard me speaking of her."
Obviously they have different means but the only real problem I have with the second sentence is the use of "speaking of" without a subject. I would also suggest "speaking about" would have been a better option. Finally I will also disregard the slight misuse, in this context, of the noun "speaking".
"You've frequently heard me speak of her."
In this case it is a generalisation you have heard me relate a story/stories about her.
"You've frequently heard me speaking of her"
of what? In this instance "of" means to relate something, you are telling a story and therefore we need to specify what you are speaking about because in this case
"speaking" is the act or skill of giving a speech at a public event:
You have frequently heard me speaking of her achievements."
speaking; noun [ U ];the act or skill of giving a speech at a public event: Ref C.E.D.
of; preposition; (RELATING TO) Ref [C.E.D.]
relate verb [T] (TELL) to tell a story or describe a series of events: Ref [C.E.D.]