"At the beginning" is more specific, which is why we usually follow it with the preposition "of" (e.g., "At the beginning of the movie, we see a man walking through New York City." "In the beginning" is not referring to the very first thing that happens, but is a more general way of saying "near the start" of something (e.g., "I enjoyed what he said in the beginning part of his speech, but then he went on for too long and I got bored.").

I think Brad nailed it. When referring to periods of time, the preposition "in" relates to open-ended periods. "In the morning, we plan to go to the beach". We associate "at" with more precise times. "Come tomorrow at noon". Of course, English always has exceptions.

They are basically the same. To the average person the meaning is the same. To get real technical you could worry about how to use the two.. With time it will come naturally. Lots of luck.

In/At the beginning , she started speaking slowly. At/In the beginning of the week, I take a bus. AFAIK, for all practical purposes, you can use them interchangeably.