What is the difference between the checks and balances system and the separation of powers system

Let's try to put it simple. The separation of powers system establishes the competencies, the actions, the prerrogatives, the domains where each branch of government can exercise their powers, different from those of other branches. For example, taxes are set by Congress, and the presidency's job is to collect them. Interpretation of laws corresponds to the judiciary, not to the president nor Congress.

On the other hand, checks and balances create counterweights to a possible abuse of power by one branch, giving branches of government legal instruments to check, to restrain the actions of another branch. For example: the US administration negotiates and sign foreign treaties, but the Senate must ratify them. Sometimes, it has refused to approve a treaty as when the Senate did not ratify the Treaty of Versailles in 1919 as requested by president Woodrow Wilson.