CN uses Daniel Berstein's EdDSA (specifically curve Ed25519). Berstein has fought the USG in court and has been outspoken about threats against our cryptography freedom.

The main problem with CN is that ring signatures require equal denominations, thus Monero has to maintain power-of-10 balances, which bloats the block chain and complicates the wallet programming.

Then when you need to recombine the change from multiple transactions, if don't mix these you break unlinkability.

Also because Monero does not force all ring mixes to be mandatory amongst preset groupings, it could be subject to combinatorial attack which could unmask the anonymity. Their researchers are studying my complaint on this issue and my suggested fix.

Confidental Transactions from Blockstream hides the values of a transaction so business privacy is retained. CN doesn't do this.

No CN coins and in fact no altcoins that I am aware of, have really solved the issue that centralization of mining can cause transactions to be censored. This is an open problem for cryptocurrency.

The other problem for all anonymous coins is that neither I2P nor Tor are reliable anonymity against a national security agency. And the nations are compiling these records to compile future tax and criminal cases against you.

(yes of course I have solutions to all of these weaknesses)