What is the difference between ‘I got to thinking about something” and ‘I got to think about something’?

Solution 1:

There are two different senses here. The more common sense (and form) is...

Once everyone had left, I got to thinking about what had happened.

...where “I got to thinking” could be replaced by “[It came about that] I started thinking”. There's no particular implication that the presence of others somehow prevented me from thinking.

But to my mind, in the less common form...

Once everyone had left, I got to think about what had happened.

... “I got to think” could reasonably be replaced by “I [finally] got [the chance to] think...”.

The first form usually means “I fell to/started thinking”, the second “I was able to think”.

Note that my examples are in the past. In the present, there's a third possible meaning...

Once everyone has left, I [have] got to think about what has happened.

In this context, “[have] got to” can be replaced by “must” (expressing present/future obligation). As Peter Shor notes below, “have” is often omitted in casual speech (in present tense; the past tense form would normally omit “got” and keep “had”).