Botox is a neuromodulator - it reduces the unnecessary contraction of muscles by blocking neurotransmitters. This reduces the pull on the overlying skin to which it attaches, thereby reducing what we call dynamic wrinkling - wrinkling associated with movement. Botox can help with static wrinkles too - fine wrinkles present at rest, but might only do so subtly. Botox, therefore, helps with the wrinkles that appear with pursing of the lips - known as barcode wrinkles.
Fillers - hyaluronic acid or otherwise - are used to create volume or sculpt. For the lips, only hyaluronic acid fillers are advisable, and they help define the margins of the lips, reduce any peri-oral wrinkles superficially, and give volume to the lips. They do not reduce muscle contractility and these dynamic wrinkles will still be visible on movement. Static wrinkles can also be improved by placing soft fillers directly under the static wrinkles.
The decision on whether to use botox or fillers depends on the issue at hand. In fact, patients typically present with a combination of dynamic wrinkling, volume loss, and static wrinkles. Combination approaches are therefore advised to get the optimal outcome.