What is the difference between them? Which one is better for what purposes?
If im writing a paper in mathematical analysis, which should i use?
There is no intrinsic difference because internally axis uses
\draw[->]... (unless you suppress the axis lines). Which one you should use depends on what you want to depict. Here are some thoughts:
pgfplotsis particularly useful if you want to plot a function that you can parametrize.
pgfplotsis based on TikZ. So if you start working with them, you may want to make yourself familiar with basic TikZ syntax first.