The answer to your question is sometimes!
In most cases when we're dealing with angles we are using the trigonometric functions, and since these are periodic in angle with period $2\pi$ it doesn't matter whether you use zero, $2\pi$ or any multiple of $2\pi$ as your equations will give the same result.
Alternatively you could be describing some object moving in a circle in an external field e.g. a gravitational field, and again most of the time tracing one circle is the same as tracing any number of circles. This is true of all conservative fields.
The exception is in electrodynamics e.g. when you're a charged object moving in a circle, because in that case you will be generating a magnetic field and each revolution of the circle puts energy into the magnetic field. In that case how many times you go round the circle does matter.