The algorithm applied (by default interior-point-algorithm, find details here. This algorithm needs the Jacobian, so either you have an analytic solution or the Jacobian will be approximated by finite differences (I guess).
Now, the number of iterations usually is lower if the analytical Jacobian is available, but to what extend strongly depends on the function itself. If the function evaluation is fast, the number of iterations doesn't matter too much. However, there are cases when each function evaluation takes a long time, say, minutes. You'll agree that fast convergence and low number of evaluations will be a huge benefit then.
If you only have to do it a few times only and you get convergence anyway, I wouldn't spent too much time in deriving the Jacobian by hand. If you have to repeat it very often, if the function has trouble to converge, or if computational time is really high, I would try to find an analytical solution. But that's just my personal experience, there might as well be other opinions.