JP to the left and US to the right.
Frame 172
Observe that the US frame is zoomed out compared to the JP frame, thus showing more of the background. This is NOT a case of the videos being unaligned, but the US version of DEMOH is actually more zoomed out. We see slightly more of the background, of the Endiness world map and of Dart. This difference in zoom is even more evident in frames 257 and 355.
The zoom discrepancy is not present in the second half of this video, which uses FMVs from various parts of the game. It is only the "original" parts of DEMOH which show a change in zoom.
It wouldn't have surprised me if the JP game had used English for the vague/non-descriptive taglines of this FMV, but as we can tell it does indeed use Japanese writing.
Frame 257
Frame 355
The game begins with Dart after a five-year journey to try and find the Black Monster, who destroyed his hometown and killed his parents when Dart was very young. Putting Dart in focus when this text shows up is almost certainly intentional, as the game has Dart postponing his search for the Black Monster and instead starts to look at the crises of the present.
When the clash of swords echo,
The journey chasing the past ends and
The journey to know today begins.
Frame 484
The JP logo has the letters appear more as though they are enveloped in fire, compared to the US (and PAL) logo which gives the letters a more golden shine. The letters have also been restored post-JP to look less damaged and fractured. When the Japanese subtitle for the game title was removed, the curve was expanded to give more of flame to it where otherwise the JP subtitle would have been. The curve now extends to behind and beyond the final letter of the logo. The background is somewhat brighter in the US version.
The space between "L" and "EGEND" in "LEGEND" is greater in the JP logo. When you see these logos side-by-side it is easy to see the US logo as an improvement, since now it looks less like it reads "L EGEND" instead of "LEGEND".
Let us leave DEMOH for a moment and look at the starting menu as it looks when you've either skipped DEMOH or watched the whole thing.
JP
US
The same changes we saw in the DEMOH frame applies to the non-FMV version of the start menu. The letters that make up the title "THE LEGEND OF DRAGOON" look more complete and restored in the US version. Additionally, the logo has been slightly moved down and to the left post-JP. Both versions have the logo text engulfed in a fire animation that can't be properly shown via single snapshots like these. I can't tell if the fire animation was changed at all between versions.
Unlike in DEMOH, both versions show just as much background so the zoom is equal. Back to the DEMOH frames.
Frame 516
Same logo differences as before. The clouds in the background start out moving at super-speed in the US movie compared to the JP one. After some time, when taglines start appearing and the background changes, the cloud animations of the US video slow down and catch up with the JP version.
The timing of the taglines popping up and disappearing sometimes match well between versions, sometimes they don't.
Frame 564
Frame 629
Frame 634
Frame 665
Frame 724
One soul seeks another and
Pledges their sworn friendship to another.
People searching for someone and then bonding with whoever they meet on the journey. Generic "adventure stuff".
Frame 805
Frame 855
Frame 884
One mystery begets the next and
The doors of fate are opened.
The Legend of Dragoon employs the "mystery behind the mystery" plot. It starts out with us wondering who the man in the black cloak is, then another mystery unveils as it is claimed that the ancient Emperor Diaz is still alive, then more mysteries and questions pop up as the journey unfolds.
Frame 1296
This frame, like the few hundreds of frames before it, uses an FMV from the actual game albeit in a different color tone. The FMV in question is "DRAGON1" which shows the Divine Dragon waking up and starting to break free from its chains (which is also relevant to the tagline). As you can tell, the zoom of this FMV shot is the same between versions.
"You are free to sever the chains of fate that bind you...."
If my memory serves me correctly then it is a point made in Disc 4 that Soa, the creator of the world, wishes for the end of humanity. The Dragoons rise up to defy this intention, which may or may not be relevant to the tagline above.