They are very similar in a lot of ways. Both Italian girls who got their start in New York City. Both relatively self-made in music - Gaga was an NYU drop out who played in piano bars, Madonna was also a university dropout who was a dancer. Both worked intensely with a producer they were also in a relationship with, to create some of their early hits. Both brought dance music back to the top of the charts, with their own unique style.
Madonna was extremely threatened by Gaga and famously called her out as "reductive" (when she meant derivative). Yes, "Born This Way" referenced "Express Yourself", but Gaga references a lot of music – she has dabbled in pretty much every genre there is. Madonna perhaps feared that her legacy would be diminished by Gaga's existence, and it has been to some degree. But Madonna has done more damage to her legacy than anyone else ever could.
They are both very different artists. Gaga has significant vocal/musical ability, and makes interesting choices in her songwriting and instrumentation. She is actually one of the most talented artists in the music industry. Madonna did create the mold for the female pop star. She was a master at garnering attention, decades before social media. Now we know she was/is just a craven narcissist, so it takes some of the mystery out of the equation. It'll be interesting to see how Gaga's career pans out, lets hope she doesn't fall into some of the traps Madonna did.