First of all, it's a "big" difference, rather than a "high" difference.
The difference between "prompt" and "push" in this context is really one of emphasis. To prompt someone [or oneself] is merely to remind them, or suggest to them, to do something. To push someone to do something is to apply some sort of pressure or coercion to them to do something. So it's easier to ignore a "prompt" than a "push".
Also, giving a "prompt" tends to be a single, one-off, action. On the other hand, pushing someone to do something can cover a series of actions over a period of time. For example, "My sister pushed me into doing this. She kept prompting me to get on with it until I gave in and did it".
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