"Go off" and "went off" have several meanings.
(see this at Merriam-Webster)
1 : explode
2 : to burst forth or break out suddenly or noisily
3 : to go forth, out, or away : leave
4 : to undergo decline or deterioration
5 : to follow the expected or desired course : proceed; the party went off well
6 : to make a characteristic noise : sound; could hear the alarm going off
The ESL Learner's Dictionary is more helpful:
(see this at the Merriam-Webster ESL Learner's Dictionary)
go off [phrasal verb]
1 a of a bomb : to explode
The building was evacuated before the bomb went off.
b of a gun : to shoot
The gun went off accidentally.
c of an alarm : to begin to make a sudden loud noise
I woke up when the alarm went off.
2 of lights, electricity, etc. : to stop working
The lights in the building suddenly went off.
3 : to leave a place for a new place
He went off to join the army after graduating from high school.
She went off to America.
4 a : to occur or happen
The meeting went off as scheduled. [=the meeting happened when it was scheduled to happen]
b : to happen in a particular way
The party went off well. [=the party was a success]
The meeting went off poorly.
5 US, informal : to begin shouting at someone in an angry way —
usually + on
Her boss went off on her because she was late again.
6 go off (someone or something) British : to stop liking (someone or
She used to like him but now she's gone off him completely.
My boss has gone off the idea.
7 chiefly British a go off with (someone) : to leave a spouse,
partner, etc., in order to live with and have a sexual relationship
He left his wife and went off with [=ran off with] some young thing.
b go off with (something) : to take (something that belongs to
someone else) away with you : steal
Someone went off with my pencil/wallet.
So, according to Merriam-Webster:
A) "She left school and went off with a group of people who were
against the war."
Sentence A means that she left a place for a new place. (ESL definition 3)
Sentence A implies that she left school for another place--to do something disreputable. (like ESL definition #7)
B) "She left school and went with a group of people who were against
Sentence B could mean that she joined a group of people who had certain beliefs--without actually going to any particular location.