The difference between Customarily and Normally

When used as adverbs, customarily means in the customary manner, whereas normally means under normal conditions or circumstances.

check bellow for the other definitions of Customarily and Normally

Customarily as an adverb (manner):

In the customary manner; as is custom
Customarily as an adverb (frequency):

Under normal circumstances, normally.

Normally as an adverb:

Under normal conditions or circumstances; usually; most of the time

Examples:

"Normally, I eat breakfast at 6am, but today, I got up late and didn't eat until 9."
Normally as an adverb:

In the expected or customary manner.

Examples:

"Lisa ate normally, until she realised that she was late for [[choir]], when she sped up."
Normally as an adverb:

To a usual or customary extent or degree.

Examples:

"He was abnormally agitated, she only normally so."
Normally as an adverb (mathematics, statistics):

In the manner of a variable with a Gaussian distribution.