Libertarians and populists are not necessarily different, but they are very different in the United States today.
Libertarians are people who believe strongly in the idea that the government should stay out of people’s lives to the greatest extent possible. This means that the government should not regulate the economy through things like setting minimum wages. It also means that the government should not involve itself in people’s moral decision-making. A true libertarian would typically advocate legalization of drugs and would support the right of same-sex couples to marry because libertarians do not want to dictate what people may or may not do with their lives.
Populists are people who believe that there is a majority of the American population that is made up of “good Americans” who need to protect the country against the elites. They feel that big businesses abuse regular American workers. They also believe that liberal elites hold values that are out of step with those of regular Americans. They want government to rein in both the big businesses and the cultural elites.
Therefore, in the US today, populism and libertarianism are poles apart. Libertarians want less government intervention. Populists, in general, want more of certain kinds of government intervention; those interventions that uphold the values of the “common people.”