Dragon Armour QuestsThe first time you wear the complete dragon armour set you learn a level of Elemental Strike, get some experience and have the temporary spell Bless cast on you. There are no special bonuses for continuing to wear it. As long as you otherwise have decent equipment, you are pretty much just missing a bit of experience by not completing the set.
There may be a way to get any missed pieces or have Grisha forge the breastplate and gem together;
Teleporting Anywhere, via hex editorYou should at least be able to pick up missed items. The quest log gets cleared when you enter the wastelands, but I do not know if that is just cosmetic, or not. Grisha may or may not fix the armour.