TIN stands for Tipo de Interés Nominal which is the “raw” interest on the loan. It excludes fees, costs of operation of the account, and differences in the amount paid due to payment frequency (such as accumulation of compound interest).
TAE stands for Tasa Anual Equivalente and is a standard defined rate that includes all fees etc. and is intended to enable comparison between loans.
Neither of them have anything to do with taxes, unless taxes are payable as part of your loan.
There is no way to “calculate” the TAE from the TIN without more information. You would need to know what fees need to be paid on the loan as well as other information about payment frequency and when payments are credited to your account.
In order to get a fair comparison you should be asking the loan provider to tell you the TAE. If they refuse that is not a good sign. The Bank of Spain even requires all banks in Spain to publish the current TAE.
The TIN is essentially a marketing tool, trying to attract you in with what looks like a low number, but which might end up costing you more in the long run.
