'force sb to do something' = You give them no choice. They 'must' do what you want... 'push sb to do something' = You strongly encourage them to do something...

Huhm... My understanding of it is like this example: "She forces him to eat the cake" ==> He doesn't want to eat the cake at all, but she demands him to do so regardless. It's kind of like she's telling him "You better eat the cake, or else ..." "She pushes him to eat the cake" ==> She may keep telling him to eat the cake, but she may not physically make him doing so.

When you force sb to do something, you want them to do it whether or not they want to. When you push sb to do something, you strongly encourage and urge them to do it, maybe even promising rewards or punishments, for example. Hope this helps :)