Hi Antonina,
In addition to the literal meanings as Olivia K. has rightly pointed out, both can also be used as phrasal verbs.
To Pull down: To reduce, bring down, destroy etc. For example, he was pulled down from his managerial position by his trusted friends.
To Push down: mostly used with price or value, to make either (price/value) decrease. For example, lack of interest in the products has pushed down the prices.
Finally, I will suggest that the two are also names of sports/exercises carried out in the gym.
Regards.