Inviter implies that the decision whether the *invité(e)(s) * will accept or not is still to be made.
-Nous vous invitons au restaurant le 26.
-Ah, désolé, nous sommes déjà pris.
Emmener implies that the decision has been made, and the invitation has been accepted (or, that there was no decision to be made, and that it was understood by all parties that the date was booked).
Ce soir, nous vous emmenons au restaurant (that implies that these persons had already agreed to spend the evening together)
BUT in this context, there is also a second meaning to inviter , which can be ambiguous: inviter can also be a polite way to mean "to commit to buy lunch/dinner". Therefore, you can even hear a guest say something like C’est moi qui t’invite at the END of the dinner, when it is time to pay the bill (and it can even be uttered by someone else than the person who made the initial invitation). In that sense Nous vous inviterons dans un grand restaurant à Paris can also be a polite way to make clear that the speaker commits to pay, even if there is no doubt that the plan is already agreed on.
Native French speaker here. In French, "Nous vous inviterons dans un grand restaurant à Paris" means "we will pay for it".
Reference: "Payer le repas, la sortie, la consommation, etc. : Bois autant que tu veux, c’est Paul qui invite." https://www.larousse.fr/dictionnaires/francais/inviter/44137
Nous vous inviterons dans un grand restaurant à Paris.
is very affirmative (You will be our guest(s) in a famous restaurant in Paris). No answer is really expected so there is almost no provision for the invited person(s) to refuse.
As this is going to happen, "We’ll take you out to a famous restaurant in Paris" is an accurate translation.
Should that have been a real invitation, the sentence might have been:
Vous êtes invité(e)(s) dans un grand restaurant à Paris.
Nous souhaitons vous inviter dans un grand restaurant à Paris.
Nous souhaiterions vous inviter dans un grand restaurant à Paris.
In these cases, you clearly have the opportunity to decline.
Finally, Larousse meaning #2 (to encourage) would be:
Nous vous invitons à aller dans un grand restaurant à Paris.
That’s the only case where the invited person(s) will likely have to pay the bill.
Using emmener in
Je vous emmène dans un grand restaurant à Paris.
doesn’t imply you are invited either, and the person saying that might also not be in the party like for example a tour driver.