Why is PSD estimated, and not simply computed?
Well, because in a lot of real world problems this


  If I have a deterministic signal with a fixed number of samples, shouldn't I be able to directly determine its spectral information?


is just not the case. Very often, measured signals are more of a random process. A simple and common case would be to have the desired signal and some additive noise, very often Gaussian in nature.

While you can capture the signal for a while and then deterministically calculate the PSD of those samples, what you get is the instantaneous PSD of these samples which may or may not be close to the actual PSD of the whole random process that you are sampling. It all boils down to what you want to do but in a lot of cases you want the PSD of the random process that you are sampling and also very often, just capturing some samples and calculating the instantaneous PSD just does not come close to it.

There is a lot of different methods for PSD estimation. Some may work better in some cases than other, maybe some are more computationally expensive than others. Or possibly some reach a better estimate with fewer samples as input data. In the end it boils down to select the most appropriate method for your given scenario.