Specifically, digital certificates can solve or significantly mitigate the problems of user-selected passwords. Users consistently choose weak passwords. Users consistently use the same password on site after site. Customer support organizations must have a way to change user passwords. Digital certificates can help resolve many of these issues.
Repudiation is another problem.
It could be a huge problem. With certs you have a legal relationship with that user. However, Marcus Ranum of Network Flight Recorder is not convinced. But for online stock trades, etc, they're relying on establishing an encrypted session and using a password over it. Certificate systems look good on paper but really won't push the envelope very much.