There are technically three distinct games that fall under the Formula Front moniker, they are...
Armored Core Formula Front (PSP)
As one of the launch titles for the PlayStation Portable the controls had to be modified (the PlayStation Portable lacked the required number of buttons for a traditional Armored Core game to function properly). As such, the game was based around using AI piloted Armored Cores. The "Formula Front" title referred to the fact that the game was a sports game set in the Armored Core world.
Armored Core Formula Front (PS2)
This was released shortly after the PlayStation Portable version and was meant as a companion to the first game. It was also the first game that featured connectability between the two devices. The PlayStation 2 version also had a limited player controlled section similar to the previous games.
Armored Core Formula Front International (PSP)
Upon the release of the original PlayStation Portable game, many complained about the inability to control the player's Armored Core directly. As such a player controlled mode was added to the International version. International was a budget release in Japan and used an English language option available in the American release. However, International featured more gaming content.
Armored Core Formula Front is not considered part of the main canon and is regarded officially as a kind of sidestory.