In modern, Western usage, they are the same game
Consider these two games:
1981 Pachisi by Whitman Games
1982 Parcheesi by Selchow & Righter (later republished by Milton Bradley)
Both of these have similar boards with only cosmetic differences. They also both use dice, are played individually rather than in teams, and have similar rules.
There are some historical differences
Bruce Whitehill published an article about Pachisi and Parcheesi in the September 2007 edition of Knucklebones games magazine. He defines some clear differences between the two:
Pachisi was designed for team play, Parcheesi for individual play
Pachisi was designed to be played by four players acting as two teams, unlike Parcheesi, in which each player plays independently; team play significantly raises the level of strategy required for the game.
The boards have a different appearance and sequence of play
The Parcheesi board still has the appearance of a cross, but the main paths have been moved to form a path around the edge of a square gameboard. Parcheesi pieces begin in individual areas in the corners of the board (one color per area), whereas the “men,” as they are often called, in Pachisi begin and end in the same center square.
Traditional Pachisi uses cowry shells, Parcheesi uses dice
One more significant difference (there are other minor ones) is that the moves in Parcheesi are governed by dice, whereas Pachisi uses cowrie (or cowry) shells.
However, most modern versions of Pachisi and Parcheesi do not have these differences, as shown above.
I suspect that the differences could be likened to regional variations of the game, or a change of the rules over time. This might be compared to Chess. Over the course of hundreds of years, the pieces on the board and the rules of movement have changed, yet the game was still "Chess". In fact, Whitehill's article lists many alternative names for the game, noting that some contain regional rule variations of Pachisi.
It seems that in this case, the rules of Pachisi have changed over time or had variations, and Parcheesi is just another name for the same game.