Weekly mowing is not just good for the overall health of your turf...it plays a very big role in weed control.
Weeds are damaged and weakened when they are mowed down. The bigger you let them grow, the stronger they get and the harder they are to kill.
This is also true for weeds you have applied a post emergent to. It can be tempting to wait and see if a post emergent is working before mowing, but that is completely counter productive. Yes the post emergent needs to dry and soak in, but after that there is no reason to skip a mow.
Mowing also helps keep weeds from spreading. Letting your weeds go will lead to them blooming and seeding, which will quickly spread throughout your lawn.
You can do all the right things...pre-emergents, post-emergents, fertilizers, aerations etc...but it will amount to nothing if you don't mow!