The definitions of these 2 terms can be found on wikipedia, but if you're interested in why I was libertarian and then became a social liberal: I was attracted to libertarianism for its minimalism and ideological purity, and therefore its [supposed] efficiency and optimality -- in libertarianism, there is minimal government involvement in the economy, and in the absence of government intervention, the market is a maelstrom of competition in which only the fittest companies offering the best goods at the best prices can survive (this was the idea, I mean). But there were 2 phenomena that led me to change my mind about an unregulated free market being optimal:

• Monopolies. I learned that the game theoretic Nash equilibrium of an unregulated market is for companies within the same business sector to merge together into a monopoly, at which point efficiency plummets and consumers lose.
• Public schools. If schools were be better off privatized, then we should find that the fittest, strongest nations in the universe have private schools, and not public schools. Instead, we find the opposite. (You can imagine the reasons why, such as human capital being squandered if public education isn't available.)

The solutions to both of these problems involve collective action in the form of the citizens acting through the government to prohibit monopolies and to provide public education. Once it was clear to me that free markets are not optimal and that some regulation and government-provided services improve national performance, the dam broke, and it instead became a question of to what extent we should have regulation and government-provided services, not whether we should have them at all.

It's worth pointing out that my "conversion" did not occur due to any kind of philosophical, ideological arguments. It occurred due to learning about game theory and due to observations of the real world. For this reason, I try to avoid making vapid philosophical arguments like "everyone should have a right to education / healthcare" when trying to persuade people who are still libertarian. Instead, I focus on the facts and evidence demonstrating that specific government-provided services improve national performance (such as per-capita wealth and life expectancy per unit cost), because that's what convinced me, and I find it most compelling.