Learn English – Is this sentence correct

Both you and your informant are correct in different circumstances.

The difference lies in the character of the verb involved. With a telic verb—one which includes an end, a specific goal or point of completion—there is a marked difference between the simple form and the progressive form: the progressive form marks the action as incomplete. Eat, fall, take are telic verbs.

With an atelic verb, which does not include an end, there is no practical difference. Stand, run, think are ordinarily atelic verbs (although in some circumstances they may be ‘recategorized’).

See the tag-wiki entry on aspect.

better for the both of us is a kind of set phrase, so this could be a witticism. You haven't provided much context, but I could easily imagine a "stand-up" comic saying it to the audience. It puts the audience in the position of the person to whom those words might be said in another context, and the conversations where that phrase is used normally have some uncomfortable element to them. It would be a kind of allusion to such an uncomfortable situation which casts the current conversation between speaker and audience in that light. I'm trying to be very general there, since the discomfort could be almost anything:

It will be better for the both of us if we stopped seeing each other.

It will be better for the both of us if you know this about me...

It will be better for the both of us if you keep quiet about all of this.