The general rule is "If the word or phrase is different, then the meaning, or at least the nuance, is different."
To push (something [on[to] someone]) informal-> to urge someone to have or buy something: "I wanted to buy a cheap washing machine but the salesman tried to push an expensive one on to me.
To deal - to buy and sell.
I agree with PaulQ that there is a difference between 'pushing' and 'dealing' generally, but in terms of casual speech drug dealers operate at the consumer level as well as the large-scale wholesale level in the US today.
In the US 'drug dealer' is much more common now than 'drug pusher,' although according to this Google Ngram the use of 'drug pusher' and 'drug dealer' in printed sources was about equal in the 1970s.