Prime Minister vs. President
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As you can see from this, the President doesn't have that much power (unless he usurps is as Bush has done); however, the PM (i.e. UK) is basically a head of state and has a lot of power and authority.
In countries that have both, their duties and powers will vary as per their constitutions. Putin is going to now make a powerful position of one that had been pretty toothless. In that instance, the power resides with the man, not the office, though.