Noise, not distance. The centre is compact enough that walking times barely separate the options. What separates them is what happens outside your window in the evening — and in a city this dense, with stone streets and shuttered windows, sound carries. A room over a piazza with bars on it is a different holiday from a room two streets back.
Whether the street empties. Florence takes an enormous number of day visitors, and the blocks closest to the cathedral are the ones most given over to them. Those streets are at their most crowded at midday and their most lifeless at nine in the evening. Areas with residents behave the opposite way.
The station, if you are moving. Santa Maria Novella sits at the western edge of the centre, within walking distance of the Duomo. If your trip involves day trips by rail or an early departure, being able to reach the platform on foot with luggage is a real and underrated advantage.
The river, and the ground. The centre is flat. The Oltrarno is flat along the river and then climbs sharply towards the hill behind it, so “in the Oltrarno” covers both an easy walk and a steep one — worth checking on a map before booking. Crossing the Arno costs you very little; the bridges are frequent and short.
Do not plan on driving. Most of the historic centre is a zona a traffico limitato — the ZTL you will see on signs and in hotel emails. It is camera-enforced, and entering it without authorization produces a fine that arrives by post weeks later. Hotels inside the zone can usually register your plate in advance, but that has to be arranged before you drive in, it is specific to that property, and the enforcement hours change. If you are renting a car for Tuscany, the sane pattern is to collect it when you leave Florence, not to bring it in.