The software that comes with the instrument (Topspin for a Bruker machine,but Varian has a similar product made by them) will certainly be able to do this, as can third party software such as MestreNova or the free ACD/NMR. You should check with the NMR service in your department as to what software they have licenses for.
The method you describe of 'subtracting' two NMR spectra, whilst possible, is not however very good in practice. For two spectra (more accurately, for two FIDs) to be subtracted, they must be IDENTICAL. That is, the peaks for aspirin in both the pure sample, and the contaminated one, must be perfectly aligned (same peak shapes, intensity, shift etc), which is practically different due to slight differences during each acquisition.
The easiest thing for you to do is probably to purify the sample, for instance by running some chromatography, or perhaps re-crystallising the aspirin, which would hopefully leave your impurity behind, which you could then analyse.
Failing this, and if you have access/inclination, some 2D NMR experiments would be useful to you. A simple COSY would allow you to see which peaks correspond to which peaks (thus allowing you to see which peaks aren't aspirin). More complex methods (HSQC, HMBC, DOSY) would also be of great help, but are probably beyond your needs for a second year lab course.