How to interpret TLC data regarding the purity of one’s sample
TLC is a great and poor method at the same time. It is great because it is quick and accurate, and it is poor because it can only show substances that can be coloured in some way and are sufficiently non-volatile to not evaporate away.

You can be rather sure of the following points:


Your impure sample does not contain any more free acid since a spot of the same $R_f$ value is missing.
Your impure sample contains a probably very high amount of aspirin, since the spot has the same $R_f$ value.


However, you cannot be sure about:


Impurities that evaporated away during spotting, development or colouring of the TLC plate
Impurities that simply don’t show up in your method of colouring (e.g. due to the lack of a benzene ring for UV fluorescence).
Solvent impurities that will never show up due to both points above.


To prove purity you would need to resort to different analytic methods such as melting point (historically), HPLC-MS or GC-MS, NMR, etc.