Power is formally defined as the amount of _work_ done over a period of time. Work is a fundamental concept in physics,  and is quantified in units of _joules_. And the derived quantity of _power_ is canonically expressed in units of _joules per second_; a _watt_ is by definition 1 _joules per second_ (it is simply defined as such by humans).

_Electrical current_ described the rate of change of _charge_ over a period of time. _Charge_ is a fundamental quantity in physics and is expressed in _coulombs_. And the derived quantity of _electrical current_ is expressed in _coulombs per second_; an _amp_ is by definition 1 _coulomb per second_ (it is simply defined as such by humans).

If you understand and/or accept the fundamental nature of what a _joule_ represents and what a _coulomb_ represents, it is self evident that the derived quantities of _power_ and _electrical current_ are fundamentally different. They are relatable things, but a third concept links them. 

That concept is _voltage_ (_electromotive force_), which is another derived quantity, and which is expressed in terms of _joules per coulomb_; and again by human definition 1 _joule per coulomb_ is called a _volt_.