gymnastic exercise vs workout

gymnastic exercise and workout both are nouns.

Word	Noun	Adjective	Verb	Adverb
gymnastic exercise	Yes	No	No	No
workout	Yes	No	No	No

As nouns, workout is a hypernym of gymnastic exercise; that is, workout is a word with a broader meaning than gymnastic exercise:

gymnastic exercise: (gymnastics) an exercise designed to develop and display strength and agility and balance (usually performed with or on some gymnastic apparatus)
workout: the activity of exerting your muscles in various ways to keep fit

Other hypernyms of gymnastic exercise include exercise, exercising, physical exercise, physical exertion.

gymnastic exercise (noun)	workout (noun)
(gymnastics) an exercise designed to develop and display strength and agility and balance (usually performed with or on some gymnastic apparatus)	the activity of exerting your muscles in various ways to keep fit
a sport that involves exercises intended to display strength and balance and agility