Trust me on this. I know what you're thinking, "Blindly trust an internet stranger? No thank you." That's good, but this time, you should listen because you care about your privacy.
It's not different enough to matter... yet.
Identifying users was still trivial as of 2 months ago, which is the last time I brushed up on implementing smart contracts. Bitcoin Lightning came out somewhere around 7 years ago. Unless something fresh and hot hit the market within the past 60 days, and has been implemented, do not, I repeat DO NOT trust your privacy to bitcoin, especially if you're doing something your governing body disapproves of.
Monero's privacy protections are still a generation ahead, but still not a big headache for the good ol' 5-Eyes. Probably not for the 14-Eyes either, but who knows what they know.