Assuming the two computers are configured identically, it is very likely the two computers are handling the certificates differently. This is frequently the case when a previous attempt has been made to access a service without the digital certificate and then with. Not to fret; this happens when caching occurs.
The certificates are probably loaded correctly so clear your cache. Tools > Options > (Browsing history) Delete > check everything > Delete.
Browse to about:blank
Tools > Options > Content tab > Clear SSL state
Close IE and try again.
You may also have a difficulty with intermediate certificates, the button next to the Clear SSL state button. You may want to delete certificates and reload them for the current user and any intermediate or root certificates that have expired.
But this is an old question that you have almost certainly worked through. Please post the answer that worked out for you.