I chose Like Both Equally. It's like comparing Apples to Oranges. Some days I really want an apple. Others, only an orange will do. They are both types of fruit, but nothing like the other.
The Beatles came first and completely changed the way American teens listened to music. They were a phenomenon. Then, a few years later, the Stones did it again, with an edge to it. Depending on my mood at any given hour, I would choose one band over the other.
Oh, and I like that you threw the Doors in there as a choice, even though they were an American born band.
For me it was not difficult. Stones are good but can't really compare (no band can) to the Beatles. I wanna be your man was a hit for Stones and was a gift from Lennon-McCartney who wrote it. I think it was their second single( Stones). Beatles later recorded it as a tune for Ringo to sing on their second (or was it third?) album.
Both bands were "Crowleyites". The Stones were more open with their Satanic leanings, such as their answer to Seargent Pepper's being Satanic Majesty's Request, and their close relationship with Kenneth Anger. Some on the more paranoid side even credit the Stones' Altamont (I think?) concert as the first professional snuff film.
I prefer the Beatles but I like to enjoy the music moreso than the men. John Lennon was a wifebeater and people in his social circles called him a phony follower who never originated any of the things he was known to have championed, rather he just parroted those around him. He also neglected his son Julian so horribly that Paul had to be the one to comfort the kid when his parents separated. John left everything, including his love, for Sean.
I used to like John the most but as I got older Paul's music was much more appealing. I don't know enough about the Paul/Faul story to say much but some of the really old pictures of Paul really don't look similar to the Paul we all know now. Googley eyes and picket-fence teeth.