Both of them mean "after". "後" is used after a verb, while "の後" is used after a noun. "朝ご飯の後に" and "食べた後" can be expressed as "朝ご飯を食べた後" and "食事の後に". Hope this helps:)

Yes, in both cases, it means "after." Sometimes, using の後に makes the sentence feel more complete than if only 後 were to be used. There isn't a strict rule that I know of, but you can just use your judgment to determine if the sentence feels alright as it is with 後, or if it feels choppy or incomplete and needs の後に.