The cost for opening Taproot channels will likely be the exact same as regular channels, as both P2WSH and P2TR outputs are the same size, so what would matter more is the inputs you use to create the channel.

Closing should be much cheaper and more private as with MuSig2 (black magic that lets you make multisigs that look just like singlesigs) it should be as cheap as just a regular Taproot spend.

Not sure if the gossip specifications for Taproot have been updated but it should reduce the size of gossip messages as well (instead of providing 2 signatures to prove you own a channel you can provide 1)

Your channel opens will look like any other taproot transaction, rather than being an obvious channel open. So a solid privacy boost. Also pay less fees to open the channel I believe, as taproot is more space efficient. 