Well those two words are depend on the conversation. I wasn't home at that time when he drop the kids off. I might go to your place at 12:00 clock so are you gonna be available at that time. Useing the word of at the time. He was already gone at the time I got there.

I can't think of any true difference in spoken conversation, besides placement. "At that time" refers to the time already established. "At the time" is when you establish the time. "At the time of the murder, Mr. Smith was at work." "Mr. Smith was at work at that time when the murder took place."