The reason you should not blow your nose too hard is that you could rupture tiny blood vessels inside your nose which can cause nose bleeds.
The nose bleeding will eventually stop though but it's best to avoid blowing your nose too hard so that you don't rupture your noses blood vessels.
Blowing your nose too hard can also send bacteria from your nose to the inner ear and cause an ear infection.
During low humidity weather conditions such as in the winter you're at risk of nose bleeds more when blowing your nose too hard and just when it's dry out.
The dry air during winter can also cause nose bleeds due to the blood vessels drying out in your nose and cracking.