They mean roughly the same thing, tittiugo. I suppose that "way of life" would be more likely in a book about cultural anthropology. "Lifestyle" would be more likely in some breezy newspaper article about the way that people in some fancy neighborhood live. However, either term could appear in either one of those sources.
Here is a graph showing that "lifestyle" came into use around 1970 and has surpassed "way of life" in frequency (in the published books that were scanned by Google Books).
My personal intuition is that "way of life" sounds like a quasi-permanent characteristic of a person, a long-term commitment,
while "lifestyle" sounds like something you can put on and take off like an article of clothing. Owlman5's word "breezy" describes it well.
"Lifestyle" is used a lot in advertising: "Buy this product to fit your lifestyle!"
This may or may not illustrate the difference, but I'll try with my own sentences:
"Not voting in elections has become a way of life for many young people under 25." = a culturally anthropologic observation by someone/long term commitments made by these young people themselves.
"Many young people under 25 years of age don't bother to vote these days because it doesn't fit in with their busy lifestyle of texting each other about clothes, sex, and music, and only knowing politicians as characters in parody comedy sketches seen on TV or the internet."
(very focused on other "breezy" inconsequential things which require neither commitment nor deep thought.)