Join both. I am a member of both and several other professional societies (AISC, NSPE, FES [Florida Engineering Society]). There are a lot of benefits from joining.
In my opinion, ASCE offers more direct involvement of its members through participation in location chapters and sections. I served as vice president and president of my chapter. It was great to see how the chapter, section and national operate. You have the opportunity serve the profession by you involvement.
I agree with Lutfi and also belong to both organizations. I would make another distinction and that is between professional societies (NSPE, ASCE, ASME, etc) and technical societies (ACI, PCI, PCA, ASTM, ANSI, NRCA etc).
While it is true that ASCE has many very well respected technical committees that produce excellent technical publications, it as well as NSPE are geared to the professional developement of the engineer members.
Concur with both Lutfi & jheidt2543 - Another benefit: In many states participation & attendance at society activites counts as professional development hours (PDH) needed annually to maintain a PE license - be sure to check appropriate state requirements.