New Orleans is a huge prize however, the possession of which could secure the British position in North America for a long, long time. Spain would realize this too and probably wouldn't be very willing to give it up at anything short of an extremely high price, like a combination of territories, colonies, and money. But this is all if Britain is being generous. If they had really wanted to turn the screws with a harsher Treaty of Paris, they could have simply demanded control of this final stretch of the Mississippi, and the Spanish, on the losing side of the war in the first place, might not have been able to do much about it. The drawback would have been an even more bitter France and Spain eager to back anyone who opposes the British, and the Prussians possibly now fearing an even stronger Great Britain with almost unshakeable control in North America and re-evaluating it's own alliances. Same goes for Austria and Russia, and the Dutch Republic would also remain opposed to the British.
British New Orleans would allow Great Britain to better integrate all of her North American colonies together, reinforcing their trade networks, and of course, virtually full control of the Mississippi River and half of it's drainage basin, including the Ohio River. There would be very little Spain could do to stop British expansion across the Mississippi, and with the increased revenues of controlling all trade and access through New Orleans, Great Britian might not end up facing the financial crunch that led her into conflict with the 13 Colonies. We might even see the British themselves attempt to create their own, earlier version of the Erie Canal to provide full lake and riverine access between the Gulf of Mexico and the Gulf of St. Lawrence. It's almost a nightmare scenario for any country that still maintains colonial ambitions in the Americas.
It's nearly a non-starter, but not impossible. The British either have to be far more aggressive and punitive in regards to the spoils of the Treaty of Paris, which likely leads to another pan-European war aimed against Great Britain, or would have to give up a lot of possessions to Spain and other parties in order to acquire New Orleans.