But even talking about the Libertarian Party, there is still an anarchist movement. Why do you say it is a different issue?

Ron Paul would be an excellent example. Andrew Napolitano would be an example of a constitutionalist, although some believe that he is an anarcho-capitalist.

Conservatives in the United States base many of their sentiments on Classical Liberalism, therefore they have some sort of alignment with libertarians. However, conservatives world-wide (true conservatism) is vastly different from liberalism and libertarianism in general.

Libertarianism has overlap with both Conservatism and liberalism.
These are in general, I know there is more to it. :)
Conservatism: Small government
Liberalism: Larger government
Libertarianism: As small of governament as possible.
By this metric they are conservative.
Conservatives: Pro-life
Liberalism: Pro-choice
Libertarians: Pro-choice
Under this metric they are liberal
There is alot more to it, but everyone would get bored with all the nuances. As well as I type at a snails pace:).

I think you're going to be surprised in the near future when there will not only be a large segement of libertarianism in the Republican party, but also the Democratic party. Do remember that until the 1930's, classical liberalism was the overwhelmingly dominant political philosophy in the United States, and it hasn't died as it had in Europe, and Asia (arguably never born in Asia.) 

Not all libertarians are pro-choice. There is an argument that it's a violation of the NAP by removing one's right to life. However, libertarians don't believe the solution is legislation. 

Because, unfortunately, a lot of modern "libertarians" are in fact just conservatives who don't want to be known as "Republican" anymore. I'm sad to say, most "libertarians" I've had the displeasure of running into/getting into any sort of political conversation with, do not in fact embody the "to each his own" type of mentality that the movement is supposed to be based around. Rather, they were, in every way, functionally republican, except perhaps wanting pot to be legalized. I realize there are libertarians who aren't like this, but I have yet to personally encounter any.

I think most of it is to induce fear.. Democrats and Liberals use Conservative to describe Libertarians because it's their way to demonize them for wanting small government.. and lower taxes.. and to not have more gun control.. etc. Liberals in turn respond to this... and as thus won't even give a Libertarian a chance.... that is mainly why there is confusion... and sometimes even hatred... (It's kind of dumb when you think about it... people actually want more taxes.. and little progress.. and more broken promises.. and the Government to spy on them... people blame Bush.. but I did not see Obama stepping up to stop it... it actually got worse under him... and he got re-elected.. )

They're not, but they seem to be more willing to ally with each other than with the likes of social democrats and other nominally centre-left ideologies.

I know:). I was just going by typical political stances. I just didn't want to get into the nuances, exceptions, ect...

basically libertarians are more similar to liberals on social issues, and more similar to conservatives on fiscal issues. There are quite a few exceptions but this is the general idea.