Batman wouldn't have become as dark and depressed after and Tim probably wouldn't have tried to become Robin immediately, although he probably would have figured out Bruce's identity.
Jason, however, did have an anger problem which possibly led to the death of Felipe Garzonas. It's likely that if he'd survived the Joker's attacks, these incidents would have continued. Maybe they'd have been lessened if Sheila Heywood had survived too, but if she'd died I imagine Jason would have been angrier than ever and still with a grudge against the Joker.
His tenure as Robin would have probably ended in disgrace and he would probably become the Red Hood once more, to get revenge on the Joker for his mother's death
That's as far as I'm willing to predict the future. Hard to say if we'd get modern Jason after all that.