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I don't know much about The Beatles and I don't know everything about The Monkees but I do know that while some fundamentals are the same for the two these bands had very different sounds from one another even when they were doing similar things
I've heard Pieces, Aquarius, Capricorn, & Jones LTD get compared to something like Magical Mystery Tour, and while yeah, there's some small similarities, I feel like The Monkees were sort of making their own stuff by that point and straying away from The Monkees+More of The Monkees and with that, they were drifting further from that "American Beatles" image (and, of course, making stuff dissimilar to The Beatles), and The Beatles were making the image a lot of people now associate with them (that you should be high as a kite to enjoy their music. Which. I've yet to try that). Even if you look at the early days of The Beatles/The Monkees, The Monkees were very much their own thing, even as a "manufactured image", they're not super Beatlesesque, they just sound like a good late 60s classic rock band.
Honestly one of the only similarities I can think of off the top of my head between the two (aside from like. Basic instruments) is the fact both bands would wear Chelsea boots. And that was extremely common in their kind of scene so like. Doesn't even matter. Also drugs but again common.
Tldr: The Monkees and The Beatles made very different music and it's weird to me that The Monkees are still (to some) considered "just Beatles rip-offs"
Edit: To clarify, I know The Monkees were made because of The Beatles. I'm not saying people compared the two with little reasoning other than the fact they were both popular 60s bands.