I wonder whether there is any difference between a last call and a final call, in the context of boarding airplanes. E.g., does one of the term indicate a higher level of urgency?
No. There's no official glossary of terms used for this purpose. Both "last call" and "final call" are simply ways of saying "we will soon leave without you unless you board."
I should add there's also no particular time frame defined universally for what "soon" means in that context. You can check your airline's policy around when doors close for more information.
The approach of this answer is based on my recollection that you are a French speaker.
French Wiktionary defines English final thus:
Its definition for English last is
- Dernier. À la fin.
(Senses 2 and 3 do not apply here.)
English Wiktionary defines last, in the relevant sense, as "final, ultimate, coming after all others of its kind," and gives the French translation as dernier. For final, it gives the definition "last; ultimate," and the French translation dernier; ultime.
The words are synonymous.
Besides, in the US, you're more likely than anything else to hear the redundant "this is the last and final boarding call for...."
The last call is the last opportunity for passengers to board the aircraft(like 5–10 minutes left for passengers to board) The final numbers have been sending to compute the final departure figures (weights, passenger count, etc.) for the flight & the final call is also to tell the passenger to take off is going & you have to come to the seat before onboarding.