Can you explain me the grammar

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Both sentences are right in terms of grammar but each has different meaning. 

"잡고 고백할 걸" means "I should have caught you and told how I felt about you (but unfortunately I didn't. Thus I regret now)"

"잡고 고백할 거에요" means "I will catch you and tell how I feel about you"

Can you explain me the grammar

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Simply put

고백할 걸 = 고백할 것인데 (hidden sentence ending: 그러지 않았다)

Should have confessed

고백할 거예요= 고백할 것이다

Will confess

(잡고 was left out because it repeats)

Examples! 

내 옛 남자친구를 잡고 고백할 걸... 난 그가 멀어지는 뒷모습만 바라보았다

I should have told my ex boyfriend my true feelings... but I just watch him walk away

저 꼭 이번 금요일에는 스펀지밥 에게 고백할 거예요

I will definitely confess my feelings to Sponge Bob this friday.

Also something to note,

잡고 or 붙들고 can often be used for exaggeration rather than being used to mean literally 'holding someone' .