(explicación en español después)
A pretext is when you try to justify why you have done something, but the reason that you give is not the real reason.
For example: the man stole the money on the pretext of being poor. (In other words, the man may not actually be poor. He stole the money for another reason.)
An excuse is when you explain why something was done, but you are also trying to lessen the blame.
For example: his excuse for stealing the money is that rich people already have too much money. (In other words, he is blaming rich people for his bad actions.)
"Pretext" significa que uno está justificando por qué ha hecho algo. Pero, la persona no está diciendo la razón verdadera.
"Excuse" significa que uno está explicando por qué ha hecho algo y quiere echar la culpa sobre otra persona o minimizar/eliminar su propia culpa.
(Yo puedo traducir mis ejemplos de arriba, si lo necesitas.)