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Botox (and Dysport) works by relaxing hyperactive muscles that cause wrinkles, such as the vertical lines between the eyebrows from scowling and the crow's feet lines by the eyes from squinting. Dermal fillers such as Restylane and Juvederm are used to fill in furrows, such as the groove that runs from the nose toward the corner of the mouth. Botox may be used for "smoker's lines" since it would relax the muscles of puckering, but it is an advanced application so you need to go to someone experienced.
Botox can be used to inject the depressor anguli oris muscle which pulls the corners of the mouth down. Botox injection would temporarily weaken this muscle and cause the corners of the mouth to come up. This should be performed by a skilled professional as injection in the wrong area can lead to a crooked smile.