One of the biggest differences is that if you are on a stipend, you will not be building up any pension benefits (and other social benefits). This makes it a lot cheaper for your employer to higher you at the same net salary.

This is not a big deal if you plan on staying in Germany only for a few years. But if you do decide to stay, the missing years of pension build means you are missing out on quite a lot of money.

(This even the case if you don't plan on retiring in Germany, as the missing years can make the difference in having reached the threshold for having build up a "vested interest" in your pension fund.)

In practice, there is very little difference in working conditions (in an academic setting) between having a full contract, and working on a stipend. 

So, if the choice is between a stipend or a full contract, with both leading to the same net income, then the full contract is (almost) always the better choice (for the employee).