A heuristic is a "rule of thumb," used because it is computationally cheap for a human brain and returns the right answer most of the time.

Analytical thinking uses heuristics, but is distinctive in ALSO using propositional logic, probabilistic reasoning, and mathematics - in other words, exceptionless, normatively correct modes of reasoning (insofar as they are done well) that explicitly state their assumptions and "show the work." So there is a real qualitative difference.