What problems does it cause if I can't breathe through my nose?

It can cause unilateral or bilateral nasal congestion, recurrent nosebleeds, recurrent sinusitis, pharyngitis, and bronchitis, facial pain, headache, postnasal drip, deterioration in sound quality, snoring, apnea, Eustachian tube obstruction, fluid accumulation in the ear.

Source http://www.kbb.org.tr