Harry could've broken his promise there's nothing legally binding about house elf promises in which case nothing much would have changed, except that dobby and harry might not have been such good friends later in the books.
Harry however does consider himself a man of his word and had he made the promise might well have kept to it. In this case a few things would have changed. Firstly harry would not have been accused of underage magic when dobby later levitates the cake, and he would have access to the letters his friends wrote. He probably would have told his friends about the whole affair, both Ron and Hermione would have then tried to convince him to go to Hogwarts anyway. If he had then things would continue much as normal.
Alternate universe in which harry is totally honour bond and not even the pleas of his best friends could make him brake his word: Ginny retakes the diary from the loo and dies releasing that portion of voldemorts soul, Hogwarts is closed therefore Sirius has a much easier time killing pettergrew. He then probably makes contact with harry.
Hogwarts would of course be closed and most witches and wizards would either go to different schools or the law would be relaxed slightly allowing for homeschooling.
Oh yeah and voldemorts soul fragment probably kills/recruits his main rival voldemort before trying to take over the world, and being beaten by harry (probably at a much later date as let's be honest teen emo voldemort has a lot more work to do than his older self to become dark lord.)
Most importantly of all vernon dursley gets a promotion at his job working with drills.
Crikey I need to go write this fan fiction now