Are doves and pigeons the same bird?

Scientifically speaking they are the same bird.

Pigeons and doves constitute the bird family Columbidae, which includes about 310 species.

Pigeons and doves are stout-bodied birds with short necks, and short slender bills (and in some species, these bills feature fleshy ceres). They primarily feed on seeds, fruits, and plants. This family occurs worldwide, but the greatest variety is in the Indomalaya and Australasia ecozones.

In general, the terms "dove" and "pigeon" are used somewhat interchangeably. Pigeon is a French word that derives from the Latin pipio, for a "peeping" chick, while dove is a Germanic word that refers to the bird's diving flight.

In real life use, we sometimes distinguish a difference:

In ornithological practice, "dove" tends to be used for smaller species and "pigeon" for larger ones, but this is in no way consistently applied, and historically, the common names for these birds involve a great deal of variation between the terms.

Pigeons tend to be seen as town dwellers. Doves more rural.
Pigeons are seen as more "dirty". Doves more peaceful.

https://en.wikipedia.org/wiki/Pigeon

Miffy expresses it very well. I just add a few helpful specifics for the more well known members of the family. The word 'dove' is often applied to the slimmer, slighter members of the family, birds of the genera Streptopelia (which includes the conjurors' "Java dove"- a white form of Barbary dove, as well as Turtle and Collared doves, and the similar Zenaida genus which includes the well known Mourning dove. Birds often referred to as 'pigeons' tend to be heavier and belong to the genus 'Columba' which includes our familiar domestic and feral pigeons and wood pigeons, though the feral pigeon is known as the Rock Dove in its wild form.

Doves and pigeons are basically interchangeable names for the same bird. There is no scientific distinction between the two. Typically, doves are smaller, with pointy tales, and pigeons are larger, with more blunt tails, but those aren't hard and fast rules.

http://www.birdwatchersgeneralstore.com/doves2.htm