Briefly means for a short time.

She appeared briefly in an Italian film.
In Britain he worked briefly as a veterinarian.
We spoke briefly about the weather.

Shortly means soon and indicates a point in time.

Shortly after her exam, she moved to Paris.
I’ll be with you shortly.

The following message should not worry you too much:

The landlord will briefly cut off electricity in the building.
This means that you will be without electricity for a short time.

However, if you get the following message, you might worry:

The landlord will shortly cut off electricity in the building.
This implies that you may not have time to prepare for the power cut.

To sum up:

Briefly tells us that something lasts for a short time.
Shortly indicates a short time before or after something.

I’ll see you briefly means that I will see you for a short time.
I’ll see you shortly means that I will see you very soon.

As I mentioned above, you can read about the words brief and short here.

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