Obviously I assume that the opposite side joining quest chain and further quests related to that side becomes unavailable, but is there something else? So far, I have done the joining the Stormcloaks quest to the point where I should take the oath. Main quest requires to retrieve the horn . Hopefully these steps didn't close anything else permanently.

All it does is affect your ability to join the other side. That is all. There are so very few changes. Painfully few.