Libertarians are against the government taking freedom away from their private lives. Hence, they tend to be for gay marriage, against restrictions on abortions, for the legalization of ALL drugs, for the legalization of prostitution, and are against any kind of censorship including any on pornography, etc. Most Libertarians are totally against the Patriot Act and find it unconstitutional. They also tend to despise any mention of religion in relation to any law. They hate someone enforcing their morals on another individual. They are also against any restrictions on immigration, except in the cases where the individuals pose a threat to national security.
Militarily, Libertarians are to the LEFT of Liberals. They are generally anti-war, and believe that military spending is way too high. An example; this years' Libertarian presidential candidate Gary Johnson said he would cut military spending by 43% in his FIRST YEAR. Ron Paul, essentially the head of the Libertarian movement, has gone as far as blaming 9/11 on our policies overseas, while also chiding the CIA for creating "mischief" internationally. Ron Paul has said that if he was elected President, he would end the IRS, the CIA, the FBI, the Federal Reserve, the Department off Homeland Security, Social Security, Medicare, Medicaid, all forms of welfare including unemployment benefits and a slew of other things. Paul also said that he would bring back ever US troop overseas immediately after being sworn in as president if he was elected. Libertarians believe in NO intervention in the affairs of other countries. I have yet to meet one Libertarian who supported the Iraq war. The majority of Libertarians I know refer to both Bush and Obama as war criminals.
So, there are many major differences between most Tea Partiers and Libertarians.