I like naruto's answer so I was going to add this as a comment, but it got too long so I am adding as answer. Consider it a really long comment with supplementary information.
When you look at the dictionary (I used kotobank), the definition of 「贅沢」 is to use more money than necessary or is appropriate for one's circumstance, and the definition for 「豪華」 literally included 「贅沢」 in its definition, so there is definitely a lot of overlap in the meaning.
However, one thing that might be useful to know is that in Japan as of now, the word 「贅沢」 no longer necessarily means "luxurious" in the sense that it is expensive. 「安い贅沢」 is very popular these days, here's an example.
You take a paid leave to go off on a 3 day weekend and stay a couple days at a ryokan with outdoor hot springs at an affordable price during off-season. It's not expensive or fancy, but sitting in the outdoor hot springs, enjoying the view, surrounded by the scent of evergreens with birds singing their mating calls while sipping sake is certainly not something you do everyday. This is not 豪華, but it would be considered 贅沢.
So 「贅沢」 is now a word that can be used to describe treating oneself to something, regardless of whether it is too expensive or not. It's more about something nice that's out of the ordinary. On the other hand, 「豪華」 is still a word you use to describe something fancy and expensive. Basically, there is a nuance difference.
Not really related, but with 「贅沢」 you can use it as a verb (贅沢をする) as well as a ～な adjective (贅沢な), while 「豪華」 can be used as a ～な adjective (豪華な) but not as a verb.