Duden tells the following
Wohnsitz: Ort, an dem jemand seine Wohnung hat
Wohnort: Ort, an dem jemand seinen Wohnsitz hat
Is it supposed to be the same now?
Wohnort has a coarser scope than Wohnsitz in my experience. Wohnort is primarily the city where someone lives, but may mean address (City+Street+etc.).
Wohnsitz on the other hand is bureaucracy-speak for address, but may also refer to the very building itself (usually upper-class, think Villas, Chalets, ..).
So the terms are very closely related, with a big shared kernel of meaning, but they also have subtle differences which makes them non-interchangeable in some contexts.
Edit: The problem with the Duden definition as replicated in the question is that two different meanings of Ort are used: For the Wohnsitz definition the "location" meaning is used, and for the Wohnort definition the "city, village, town" meaning is used. So one could actually resolve Wohnort to "City/Village/Town where someone's (location where someone's dwelling is) is".
There are two differences between Wohnort (literal translation: 'locality of living') and Wohnsitz (literal translation: 'seat of living'). The first concerns the level of detail implied:
If you just want to indicate the Wohnort while using the word Wohnsitz, the most standard way is using the preposition in: "Dagobert Duck hat seinen Wohnsitz in Entenhausen." Some people probably drop it. That's non-standard, but much less so than including a street address under Wohnort would be.
The second difference is that, unlike the plain and normal word Wohnort, Wohnsitz is primarily a bureaucratic term that has a very precise definition. Or more precisely one in Germany and a similar one in Austria. (I don't know about the situation in Switzerland.) A person can have several Wohnsitze, in which case one of them is the Hauptwohnsitz and the others (rarely more than one) are Nebenwohnsitze. This should become clearer from the following discussion:
(Read the following details only if you love absurd bureaucracy or if you are planning to live in a German-speaking country.)
Your Wohnsitz is the address where you live - regularly and in a non-temporary manner, and where you consequently have to register with the authorities so that the state always knows where to look for you. (In case you owe someone money, or they want to deport you, or whatever.) In principle everybody must have one. It's OK for homeless people to register a place where they don't actually live as Wohnsitz, so long as the person or organisation there agrees, they can usually be contacted via that address, and it's at the locality where they spend most of the year. This is important because a Wohnsitz is a prerequisite for a bank account (with it, banks have to give you one), and a bank account is a prerequisite for certain social benefits. Also, a German living in South America or Australia and returning once a year for a week to stay with his or her parents, may still have a Wohnsitz there. As it is the only one, it will be the Hauptwohnsitz. Austrian bureaucracy has a sligthtly different point of view that takes into account other countries. Consequently, if our hypothetical German also spends 2 months every year in Austria with his girlfriend and another 2 months at the other end of Austria with her and her parents, then in German bureaucracy he will have a Hauptwohnsitz in Germany (and other countries don't matter), and in Austrian bureaucracy he will have a Hauptwohnsitz on another continent and possibly 2 Nebenwohnsitze in Austria.
I am not completely sure what happens when someone has more German (or Austrian) Wohnsitze than Wohnorte, e.g. lives at two addresses at opposite ends of Berlin. I think most bureaucrats don't really care about this so long as you don't try to argue that you couldn't attend your trial for treason because the letter went to the other end of the city, where you haven't been in months even though it's formally your Hauptwohnsitz. Since the choice of Hauptwohnsitz may have a financial effect for the citizen and for the localities involved, I would assume that there are rules designed to prevent splitting the time you live at the same Wohnort throughout the year (which is often relatively easy to prove if you have a normal job) evenly over so many addresses that the few weeks you spend in some remote location every year are enough to make your only address there your Hauptwohnsitz.
But most people have a city where they actually live most of the year - their Wohnort -, and their only address in that city, which according to law is their (Haupt-)Wohnsitz, has in fact been registered as such.
Der Wohnort, wohnhaft, die Wohnadresse, und das Domizil einer Person sind geografische Ortsbestimmungen des Wohnsitzes. Der Wohnsitz ist der juristische Bestimmungsort einer dort wohnenden und lebenden Person. Er braucht nicht mit dem Wohnort etc. identisch zu sein sondern wird durch das Recht bestimmt. z.B. ein deutscher Wohnsitz ist die Wohnstätte eines Deutschen in Deutschland und auch im Ausland solange dieser die Staatsangehörigkeit des Landes seines Wohnortes nicht erworben hat. (Ein deutscher Wohnsitz in Italien ist also kein rechtlicher italienischer Wohnsitz). Im Schweizerischen ZGB Art. 22 wird der Wohnsitz einer Person genau bestimmt. Da wird genau bestimmt wann sich der Wohnsitz rechtlich ändert.