Is the game different in any way if I chose Jude?

When i started the game for the first time i was given a choice between Milla and Jude (which i assume is the game skipping the title screen and going to New Game) and i chose Milla.

the game then showed Milla and Jude encountering one another just outside the sewer entrance to the lab. further in just before the Lilium Orb tutorial i meet Jude and it seemed there was a bit of story that was skipped over (something about a professor who called Jude to the lab and apparently they're dead).

I am wondering, if i started a new game and chose Jude, would the game be any different as opposed to me selecting him as the party leader after choosing Milla?

There are a few sections that differ between Jude and Milla. In general, Jude gets a bit more content but Milla is more plot important. In general, you get the impression that Jude was meant to be played first, with Milla adding a bit more detail after you have already finished the game once. That said, you only learn one engimatic party member's motivations in Milla's story (this lack of explained motivation REALLY bothered me on my playthrough as Jude)

In combat, you can play as whichever character you want regardless of which character you pick as the protagonist.

It's important to note that the content is the same 95% of the time. Feel free to start a new game as Jude just to play his intro, because his intro does have a lot more content than Milla's intro. After that, though, just play whoever -- I don't think that playing as one or the other will affect your enjoyment of the game.