One of the functions of と is to highlight a relationship between a noun phrase with the subject of a clause. This is often used in comparisons. It's similar to the English word 'as' in this sense. For example:
この車は僕のと同じです。This car is the same as my car.
Here, the と serves to show that the subject (この車) has a relationship with the other noun phrase (僕の車). That relationship is then clarified (同じです). It's like saying
the relationship between Noun A and Noun B is X.
Similarly, in your sentence 今と変わらない生活, the subject is inferred as the lifestyle of 私はお金持ちになった and と indicates a relationship between that and 今. This relationship is then clarified as 変らない. In other words,
the relationship between 'rich lifestyle' and 'current lifestyle' is 'nothing different'.