First off, I don't care much for either of these parties (I usually agree best with Green candidates, though I'm still a true independent), but I do think Libertarians are better than Democrats. They stand for particular issues, while Democrats rally around the opinions of whoever happens to be most popular in the party at any given time. Libertarians primarily value individual freedom, while Democrats are willing to crush any individual in favor of popular opinion. Libertarians may advocate a weaker government, and some even no government at all, but that's still probably better than a grossly mismanaged one, which is what both of the major national parties stand for in their own ways (both Republicans and Democrats). Democrats gave the White House to Obama, who may have done a few good things, but has also been responsible for many more ridiculous, corrupt, or even evil ones. To name the three worst: He endorses mining for natural gas and oil on what were people's farms, for no reason other than corporate interests and flimsy sub-surface mining rights deeds that were never written with this intention. He's openly given himself the power to arbitrarily imprison any individual U.S. citizen for as long he wants. Worst of all, he's given himself full and unchecked power to arbitrarily assassinate any U.S. citizen for any reason he wants, whether publicly announced or not. No one individual should have the right to arbitrarily kill in a democracy. That's simply disgusting. It also seems likely that Democrats will endorse Hillary Clinton for the presidency in 2016. She doesn't even believe in freedom of speech, separation of church and state, or a consistent healthcare system (either single-payer or free). Libertarians may never be willing to institute true universal healthcare, but neither is Hillary. She wants "American Health Choices", something even worse than not being given the universal healthcare we deserve, or having to live under Obamacare. Libertarians may not be willing to institute any strong government policies, but that's still better than living under the corrupt ones advocated by Democrats.
My previous response wasn't all that refined, since it was written quickly, but those are my basic arguments. One clarification: Hillary may not necessarily be opposed to several of the basic freedoms valued by Libertarians, but just doesn't seem to support them as strongly. Something to add might be honesty--Democratic politicians seem to lie or make ambiguous statements more often than Libertarians.