This game has seen a few different versions since its original 1991 release:

Game Boy
(Japan)

This game was first released in Japan, using the name Dracula Densetsu (Legend of Dracula). It was released on the 27th of October, 1989.

Amusingly (to me) this game's storyline eventually led to problems when Castlevania III was released for the NES, check this out at the Castlevania wikia for more info.

Game Boy
(America)

The American version of the game was released on the 6th of December, 1989. The title screen was changed, with the background graphics removed and a new logo put it. Other than this however the game is exactly the same as the Japanese version.

It is never mentioned anywhere in the game the name of the character you are playing as. The manual and back of the box also fail to mention it. However, the back of the box makes this statement:

"After getting his wings clipped in Castlevania and Simon's Quest, the death defying Count has risen again."

This makes it sound like the game is a sequel to the NES games, and you are playing as Simon Belmont. In fact, in some gaming magazines, it was stated that you actually were playing at Simon, which isn't the case. Things would be cleared up once the sequel was released.

Game Boy
(Europe)

The game was not released in Europe until the 8th of August, 1991. Other than some minor changes to the title screen, this version is the same as the American version.

Game Boy
(Japan)

The game's first re-release happened on the 25th of September 1997, when it was included as one of the games on Konami GB Collection Vol. 1. This collection was only released in Japan. The collection's intro features a small scene showing pixel art based on the original version's box art, and a short extract of the first stage's music plays during this. This version also has Super Game Boy support, featuring a new image of Christopher Belmont, alongside a castle.

The biggest change in this version if that Christopher moves faster than he did in previous versions -making the game a bit less frustrating in certain sections.

This version also features a character from Tokimeki Memorial, and there is a extra mode on the main menu where she can give you advice about each game on the collection.

Game Boy Color
(Europe)

In the year 2000, Konami GB Collection Vol. 1 was released in Europe, and The Castlevania Adventure is included on it. This version adds full color to the game if you play it on a Game Boy Color, although it also works on an original Game Boy system. It features the same intro as the Japanese collection version, although the title was changed to just "Castlevania". The Super Game Boy support featured in the Japanese collection version was removed, along with the Tokimeki Memorial character and advice mode. The title screen was updated, restoring the moon and bats originally featured in the Japanese version, although strangely the copyright text was removed. Christopher's faster walking speed seen in the Japanese collection is also featured here.

Nintendo 3DS
(Japan, America, Europe)

This game was released on the 3DS as a downloadable Virtual Console title. It's the same game as its original release, other than having one very important new feature: savestate support. You can save the game at any point, which is obviously incredibly useful.