Would it be really difficult to get a good night's sleep in one stayed in the Vieille Ville in Nice?

I stayed at a hotel in the old city and didn't have a big problem with street noise. It was on a pedestrian only street with bars and restaurants and shops. Now it may just be that the hotel was well soundproofed or that my room just wasn't right over the street (I don't really remember where the room was) but all was fine. Not sure where the apartment is that you are looking at or what type of building it is. Are there any reviews of that apartment for you to read? If noise is a problem you can be sure there will be something in the reviews about it.

By the way, I loved staying in Nice and didn't have any issues with crowds or traffic, but I wasn't driving.

If you are at all noise-sensitive, I advise traveling with, and using, earplugs. I live on an urban street corner, so I'm used to a certain degree of noise, but I still use earplugs in Europe except on nights when I really, really, have to be sure I hear my smartphone alarm.

A friend rented an apartment for six months close by the center of Nice. The street noise was fine, but the people in the next apartment happened to be very noisy. So you never know.

If you're really not a city person (crowds, traffic, noise) then also consider that a rental in a building with other short-term / holiday rental units is going to have noise, perhaps, from the other occupants.

I stayed in an AirBnB listing a few blocks from the Cours Selaya market on the top floor and learned to keep the skylight closed not because of noise but because of mosquitos. The building was on a street highlighted in the RS guide map for its bus stops on the relevant routes for travelers. I don't recall much about street noise, but I do recall being able to hear the comings and goings of the other occupants -- the exterior door, the interior door, and the stairwell doors were never all shut and secure at the same time; there was one ajar or blocked open all the time. So I made a point of locking the apartment door. (At well-run B&Bs I sometimes don't see the need to lock the bedroom door.)

In Paris I stayed in an older apartment unit in the 18th arrondissement. The floors and ceilings were wooden, so that there really wasn't any soundproofing, and the woman's constantly walking around was like hearing someone stomping. Strangely enough, I didn't hear sound from adjacent units, maybe because of the way the building was divided into a lozenge with a courtyard in the center.

I could hear people in the stairwell and coming and leaving their apartments frequently, but I had my blutooth speaker muffled well those loud sounds of banging doors. But people did not talk loudly in the hallways, and I rarely heard conversations from other apartments, which was very nice.

The Vieille Ville in Nice I understand has many bars, restaurants, and even discos, as well as steady traffic from shoppers and tourists, so that is an additional source of (unwanted) noise.

I stayed in Villa Leonie in Nice, and it was on a busy street, but it had good windows, and the noise was not an issue. I am a light sleeper, too.

I think that Antibes would be kind of noisy near the Old Town as well, as it is a resort town with lots of bars boutiques and restaurants. But it would be smaller...I always take several pairs of earplugs whenever I travel, just to ensure sleep. But being in an apartment where there one can hear loud laughter and conversations (from the street or from neighbors) can rob one of enjoyment. A certain amount of noise, of course, is inevitable.

Buildings in the old quarters of town are very close together, and the streets are narrow.
There will be garbage and recycling pickups nearly every day.
You will hear the trucks anywhere from around 6 AM until the early afternoon.
If you stay anywhere near a restaurant or cafe, there will be deliveries, normally around 7AM until 5 PM.

If you are really this sensitive to noise, perhaps it would be better to stay on the outskirts instead of the old quarters, where you'll probably be miserable.

That is why staying too close to the center of Nice or even a smaller city/town like Antibes doesn't make sense for someone sensitive to noise.