Видаетесь means see one another habitually, on a regular basis, which is suggested by its imperfective aspect. BTW this form is obsolete. Today we'd say видитесь.
Увидитесь in this context means will see one another soon, in a short while, next time.
I can think of 2 justifications for their combined use:
1) they denote different types of action
2) purely literary aesthetic considerations.
The reason for use of видаетесь instead of видитесь (provided the latter form was current at the time of the text composition) could be that since the latter is consonant with увидитесь thereby creating a semblance of tautology, which is frowned upon in the ethos of Russian composition, it was discarded in favor of an alternative form for the sake of euphony.