TLDR; Monero privacy is better

The good: 
Lightning *sender* privacy is good (onion routed) and it is ephemeral (must actively probe and collect to get any data at the time transaction occurs or it's gone)

The bad: 
Lightning *reciever* privacy sucks, your IP is not hidden by default, and even hidden balances can be discovered. 
"We identified 27,183 private channels, discovered hidden balances, and showed how a passive adversary can infer payment endpoints with very high probability."
https://arxiv.org/pdf/2003.12470.pdf

For Monero, IP address is hidden by default (Dandelion++), and amounts and addresses ARE NOT EVEN VISIBLE! (using Confidential Transactions and Stealth Addresses) They are completely hidden!

I would strongly recommend Monero if privacy is priority. Although lightning is an improvement over on chain privacy.