I use both; a small variety of combs and brushes, plus fingers.
My current technique takes care of all my training, styling and conditioning maintenance needs and is, as follows:
(also, bear in mind that this works for me because I have trained my 'tache for the better part of a decade, and it's long; this may not work for a petite handlebar wearer and someone using styling products).
Towel dry and pull damp whiskers to sides with
fingers;
apply (LUSH Retread) conditioner to graspable extremities only (hirsute, and of the upper lip variety) with
fingers, and leave in;
load
comb with a leave-in spray conditioner and/or heat protection spray and
comb through entire 'tache, only if required;
directional training with detangling
comb/brush and blow dryer (currently, a lower heat setting than once used);
shape and create upward curve with
fingers, while pulling end whiskers outward to side; now substantially dry,
begin to style with bristle brushes - initially curl around small round
brush and hold briefly under blow dryer to 'set' curve or curl; then,
a (finishing) boar bristle
brush and fine tooth moustache
comb for directional finish styling and any final light stylistic touches, respectively;
no styling products used and I walk away most days in about two minutes, five maximum on a troublesome day.
[To be honest, most days the foregoing can and is done with the fingers only]
During the day I use
fingers to sweep bottom whiskers to sides and away from mouth. I also carry a small moustache
comb for any critical touch-up, should a real need arise.
Prior to bedtime I return to yet another boar bristle
brush and spread the natural oils throughout the whiskers;
I read one page of a book and fall asleep

;
awake each morning to find my 'tache completely dishevelled by the pillow; and,
the whole process begins again;
or not, depending on what the man in the mirror says.