For one, no exits until you exit out of the thing.
I don't see that as a big issue- it's not like the existing Columbia/Seneca ramps are hugely used and the current viaduct doesn't have much access into downtown. The vast majority of traffic is Port of Seattle/West Seattle Bridge traffic, which would still have access to the tunnel.
For another, bad, New York-like congestion when construction actually takes place.
Yeah, it's bad for a couple years, but some major construction is going to have to take place, even if it means building a new viaduct. The current structure would have to come down before they could build a new viaduct, which is really the only other option (assuming the boulevard option has been struck down, but that was a horrible option to being with).
I don't see much value in keeping the waterfront separated from downtown just to save two ramps, which is all you get by keeping the viaduct. I admit I've used the southbound one off Columbia after Mariners games to head to Tacoma. If you're parked in the Kingdome lot it's much faster to go up First and then take the viaduct back down, but beyond that there's not much value. If I ever do that again, I'll just park south of the stadium. There's already no way to get off the southbound viaduct into downtown and no way to get on the northbound viaduct, so it's not as if a tunnel really restricts things.