This is a tricky one, and as is often the case with languages, there are people who argue that the two words can indeed be used as synonyms. However, according to the majority of sources I consulted, and in line with my gut feeling of the English language, briefly and shortly are not interchangeable, and indeed can have a very different meaning. Let us first consider the adjectives brief and short, which state that whatever they are describing lasts only for a short period of time. For example,
He will pay us a short visit.
is equivalent to
He will pay us a brief visit.
However, the adverbs briefly and shortly have a rather different meaning. The sentence
He will come over briefly.
means that he will come over and stay for a short time, whereas the sentence
He will come over shortly.
implies that he will come over very soon and stay for an unspecified amount of time.
As you can see, using the wrong word can completely change the meaning. Depending on the context, that may have dramatic consequences. The warning
We will briefly switch off the internet.
sounds rather harmless, whereas
We will shortly switch off the internet
will make people much more nervous!