Lawyers are trained to argue both sides of any issue, so I'm not surprised they did. As usual, the facts of the case can tip a court ruling either way.
When two constitutional rights collide, courts can go either direction or split the difference based on the facts of the case.
The first amendment isn't absolute in the sense that perjury, fraud, slander, libel, copyright violation, trademark infringement, invasion of privacy, outing spies, verbal harassment, and many other forms of expression remain illegal under some circumstances.
Similarly, the right to privacy is by no means absolute. As public figures, Biden and Palin legally have lesser privacy protections than private citizens, in service of first amendment rights. Yet, many rights, including abortion rights, are predicated upon the implied constitutional right to privacy, so courts are hesitant to set precedent that weakens privacy rights, even for public figures.
To build a case against Gawker.com, many questions must be answered. Is the published information true (if not, libel is possible, but not likely)? Was the info obtained illegally? Did Gawker "materially participate" in any illegal act (pay for screen shots or enter the hacked password)? Would the info have been legally available anyway (under Freedom of Information, or the inherent openness of unencrypted e-mail)? Is there evidence of harm (did poll numbers drop after the revelation, or was fraud or harassment committed as the result of publishing the info)? Is there evidence of malice (a key Gawker employee bad-mouthing the victim)?
In this case, the suspects allegedly obtained the account's password in violation of all kinds of information security legislation, perhaps most infamously the DMCA. If Gawker.com then logged in with that password to make its own screen shots, it would probably be found as culpable as the original hackers. Gawker would be more difficult to prosecute if it merely redistributed screen shots without directly breaching the account, so claims must be verified.
To me, though, the central question is whether or not e-mail, in general, even allows for a legal "expectation of privacy". Judges have answered both "yes" and "no". Most e-mail users will answer "yes, even if unencrypted", while most network geeks will answer "no, not unless it's encrypted". Without encryption (like OpenPGP), it's relatively trivial to intercept e-mail with a packet sniffer, which is necessary to maintain network security. If e-mail info is inherently public, then Gawker.com has only republished info that was already public, which is not a crime, even if crimes were committed to obtain the public information.
The precedent with respect to public figures is mixed, but generally, it is illegal to access their personal e-mail without permission or a court order except for unavoidable maintenance purposes. It isn't a stretch to claim the e-mail is thus "private information", so publishing it without permission can be illegal. Naturally, the line between public and personal email can get blurred for public persons.
Short answer is, I got no clue.