Learn Russian: Они купили этот дорогой замок только ради престижа и тщеславия. - They bought this expensive castle only for the sake of prestige and vanity.

You could, and it would be grammatically correct (для престижа), but ради is much stronger. While для just means 'for', ради translates to 'for the sake of' or 'in the name of'. It specifically highlights the ultimate driving motive behind their expensive purchase.

Why do престиж and тщеславие end in -а and -я here?

They are in the Genitive case, which is required by the preposition ради. The masculine noun престиж takes the -а ending, and the neuter noun тщеславие changes its final -е to -я.

The word тщеславия is quite long. Is there an easy way to remember it?

Yes, looking at its roots helps! The dictionary form тщеславие (vanity) is a compound word made from тщетный (vain or futile) and слава (glory). Literally, it means 'futile glory'.

Since этот дорогой замок is the direct object being bought, why doesn't its ending change?