Austrian economics is a term used for free market economics. It is named so not because Austria had or has a free market economy. It is named so because the people who originally studied and advocated it were from Austria. The best place to read up on Austrian economics is the Mises Institute, which is named after Ludwig von Mises. Although there were some that came before him, Mises is really the epitome of Austrian economics.
So what is the difference between Austrian economics and libertarianism? Can you advocate Austrian economics without being a libertarian? Can you be a libertarian without advocating Austrian economics?
There is a difference between the two, besides the obvious. Obviously Austrian economics just deals with economics, while libertarianism deals with all issues political. But the differences do go beyond that.
First, there are some people who call themselves libertarians, but they don't necessarily follow Austrian economics. It could be contested whether these people are actually libertarians, but they are going to call themselves whatever they want. For example, there are followers of Milton Friedman who would describe themselves as libertarians, but these people obviously aren't the radical libertarians that would be more likely to follow Austrian economics.
But there is another point to be made here. It is possible to be a follower of Austrian economics without being a libertarian. It is a strange thing to think because it is so rare. I have seen this pointed out before, but don't have a link available right now. I believe it may have been Walter Block who made this point.
With libertarianism, it is really just an advocation of what the law should be. If you are a libertarian, you think there should be laws against murder (whether this is a government law or private law). You think that laws should prohibit the initiation of force and perhaps enforce contracts. You don't think there should be laws which redistribute wealth or make things a crime where there is no victim.
Austrian economics is the study of free market economics and uses human action as its basis. Most followers of Austrianism would advocate laws permitting a free market environment. But you could technically be a follower of Austrianism and still advocate statist policies. You could have a politician who understands the Austrian Business Cycle Theory and also understands the other evils of monetary inflation. Yet, he might care more about his job and pleasing his constituents and advocate statist policies. He understands that these policies are not good for most people (although he probably wouldn't say that), but he acts in his own short-term self-interest by advocating bad policies.
I see morality playing a big difference between the two. Libertarianism really involves moral principles. You can certainly advocate utilitarian outcomes with libertarian policy, but it tends to be backed by moral arguments as well. While many Austrians will, for example, call taxation theft, it really does not go to the heart of Austrian economics and is more of a libertarian argument. Austrian economics tells us that humans act freely and that government interference in the marketplace will likely lead to bad and unintended consequences for the general populace.
So while these two terms overlap, there can be a difference. You could have someone study Austrian economics to help them with their investing and yet they may not care one iota about living in a free society.