I keep thinking about this one and it is tricky.
The only way I can think of to differentiate between the two is that when you do it 'on' something, it is a much more precise and smaller area/thing...but even that doesn't always help. I think it is just one of those things you have to get a feel for.
Push the car. No, not there, push on the back of it.
Ok, we'll pull the car to that house. Can you pull on the gate to open it for us?
Oh dear, suck the splinter out of your finger. If you suck on your finger hard enough it should come out.
That dog is trying to bite me! It's biting at my leg as I go past. Ouch!
Here, bite on this bit of wood to help you stand the pain.
Often they are interchangeable. Pull the rope to ring the bell. Pull on the rope to ring the bell.
'Bite on' is a little easier. It means to hold something tight between your teeth without necessarily biting into it. Oh dear, there's another option, into.
I give up, sorry.![Sad [:(]](/emoticons/emotion-6.gif)