(Full disclosure: I'm an HLS 1L)
The bottom of the bottom of HLS can get biglaw job offers. That isn't as true at CLS, but assuming you have at least a decent grades, you're probably going to get offers if you're at CLS too.
In my experience, everyone with a pulse at CLS can get a biglaw job offer. I don't think anyone seriously worries about being employed on graduating
If you don't want to do big firm law, I think HLS wins hands down. We have a great public interest placement office. I had opportunities to do whatever I wanted this summer: public interest in Ireland, Russia, South Africa, urban or rural America. Literally, whatever I wanted. Also, you need to remember all public interest jobs aren't created equal. The more important ones, the ones where you're getting on the track to eventually argue important cases with nationwide implications, are difficult to get. Anecdotally, I've seen the HLS name alone (we don't even have grades yet) help a great deal in landing these sorts of jobs.
Again, I don't think there is a significant difference between the two schools in this area. Columbia guarantees summer funding for public interest work both 1L and 2L summers (I would imagine Harvard does the same), and the job options are phenomenal. It's hard to imagine that any other school could do much better than Columbia in terms of international job opportunities. I spent my first summer doing a human rights project in Pakistan, and knew people working in London, the Hague, Serbia, India, Tanzania, rural West Virginia etc.
But this doesn't capture one of the best parts of Columbia, and what in my opinion makes it arguably the best law school experience in the country. We don't have to wait until the summer to get involved in the most interesting and important work! Our placement in New York means that the school is connected to all sorts of organizations large and small, national and international. Students are doing cool projects and making connections all year round. Columbia has official externship and clinical programs where you can work with a judge in the 2d Circuit or the Southern District of New York (almost a part-time clerkship while you're still in law school!), the New York U.S. Attorney's Office, or do projects with the UN, the ACLU (their headquarters are just a subway ride away), various NGOs and community development organizations, and many others. And the alumni network in the city is so strong that you can find a ton of other, unofficial opportunities while you're in school. I know someone who is interested in intellectual property, so he connected with an adjunct professor and is working on the Viacom/Youtube litigation, and that's not a particularly unusual story here.
NYU is also in the city, but they don't have the same range of programs and opportunities. And there is no substitute for many of these opportunities if you're not here, because this is where so much of the action in the legal world is centered.
In my opinion, Harvard offers a small advantage in prestige, especially among non-lawyers, but not a corresponding tangible advantage in the opportunities you have after law school. And I think no school matches Columbia in terms of the incredible educational and experiential opportunities you can take advantage of while you are in law school.