Yes there's a difference. I understand your question is about the difference between 'to push' and 'to push against'. If you 'push' something then normally that thing moves. If that thing resists movement, even temporarily, then you 'push against' it. So for your example sentences we need to consider that a wall is normally a permanent structure that won't move. Therefore normally you can 'push against' a wall with all your strength and it will not move. In what context can we say 'He pushed the wall'? Only if there was movement. We'd have to imagine a strange context, but it's a grammatically valid sentence. In practice we often choose a phrasal verb like 'push over' , 'push back', etc to express the type of movement involved