Budva is one of the oldest towns in the Adriatic, originally a Phoenician settlement, the Greeks and Romans have also played their part as have the Venetians and Serbs. In 1944, like much of Montenegro and Croatia - Budva gained independence (as Yugoslavia) from fascist Italy.
In 1979 it suffered a disastrous earthquake but I didn't actually find out about that until after I'd been there and there's definitely no sign of it these days. The first time I visited, I stayed in a rented room from a family which were advertising "sobe" outside of their house. This cost 6 euros per person but will undoubtedly be more expensive now. The last time I visited, I was staying in Kotor.
The old town of Budva is undoubtedly cute with narrow cobbled streets, nice architecture and a fort but it doesn't really appeal to me, the streets are always filled with hordes of tourists following each other with postcards and ice cream in hand. Whilst the beach has fine sand and there are plenty of good cheap eateries around, the strip is so packed that it sometimes feels as though it's the only beach on the Adriatic. Rich millionaires from Serbia and Russia use it as their personal playground whilst their also loads of expats from Ireland and UK who have invested their due to the property boom. Budva could be Varna in Bulgaria or Praia da Rocha in Portugal, any resort with very little character.
I'm sure it wasn't intended that way but that's what has happened over the years! It is worth visiting though when in Montenegro simply for the use of the beach.
I wouldn't like to stay there again though. Nearby Sveti Stefan has an interesting history and used to be a fishing village but it's now just a single resort which is often used by celebrities.
This part of Montenegro is the most popular and by far the least interesting for me. I'd recommend Kotor, Herceg Novi, Niksic, Cetinje and Skadarsko Jezero over Budva any time. I honestly can't see why people would go all the way to Montenegro to go to a resort style place which could be found anywhere in the world.