There are already more thorough guides that cover the numerous visual and gameplay differences between the two versions, so I'm going to just list a quick summary of how they affect speedrunning here.

Character differences:

Alucard

Bat form depletes your hearts twice as fast in the USA version. Since Alucard moves faster in bat form, this is a disadvantage for the USA version.

Grant

In the Japanese version, his regular attack is a throwing dagger (just like the dagger subweapon), meaning he only gets the axe and stopwatch as sub-weapons. In the USA version he can have the dagger, axe, or stopwatch as a subweapon, and his normal attack is a close-range knife stab. It does the same amount of damage as the throwing dagger, but it has a quick repeat rate, similar to Sypha's staff. This ends up making the USA grant speedrun quite a bit faster.

Grant's wrong warp glitch also only works in the USA version, which is why all the Any% runs are USA.

Sypha

The only difference is her lightning orb subweapon -- in the Japanese version if you hit an enemy close range, all three orbs will hit, doing massive damage. It also moves faster and homes a bit better. In the USA version it's nerfed such that you'll only get credit for one orb hit if you're too close. This lets the Japanese Sypha get some quicker boss kills than are possible in the USA version.

Trevor

His name is Ralph in the Japanese version, which is completely awesome.

General differences

• Damage is calculated very differently in the two versions, drastically changing how and when you damage boost. Typically in the USA version the damage you take is mostly a function of what stage you're on -- early stages have Trevor taking two bars of damage per hit, progressing to four by the end (Grant & Sypha always take an extra bar of damage per hit). There are a few exceptions such as the Cyclops' hammer and the Leviathan's fireballs. In the Japanese version the damage is consistent throughout the game, and is based on which enemy and/or projectile hit you.
• Some of the candles drop different items, and some candles are in different places.
• Bone pillars take more hits in the USA version, and their fireballs move faster.
• The mummies' projectiles act differently in the USA version (they fly in a wavy pattern instead of straight). This doesn't matter for most routes since you tend to just vaporize them with holy water, but I imagine it could affect for ex. the Japanese Grant.
• In the USA version, dying to Dracula spawns you earlier in the stage than the Japanese version. Of course you don't want to die in a speedrun anyway, but it does make recovery slower and more difficult.
• The cross and holy water do not ¤typically¤ appears as random drops in the USA version. I have seen it happen, but it is extremely rare -- I suspect it is some sort of yet-unexplained glitch as opposed to an intentionally low probability drop.
• Dracula's 3rd form lasers are much shorter in the Japanese version, and don't shoot upwards. This makes them far easier to dodge.
• Dracula's 1st form teleport is slower in the USA version.
• The second skeleton knight boss that you encounter on Alucard's route gets a lot more flying bones encircling him in the USA version. This means killing him with Alucard is too laggy -- you have to switch back to Trevor and use the holy water.
• There are other differences in enemy spawn locations and behavior that I'm not catching here. In general the USA version added more bats and various other things to make the game tougher.

2nd Quest Differences:

In the USA version, some new enemies appear:

• The medusa heads are replaced with flying skulls, which have faster and much more random movement.
• The spear knights are replaced with, well, a different spear knight. The new ones look different, take more hits to kill, move faster, and actually stab with their spears.
• The zombies in stage 8 are replaced with some kinda robed ghoul thingies. They move faster and take more hits to kill.

Also...

(???) The water dragon boss in block 6D has more random spawn locations in the USA version. They may be random in the J version as well but I've only seen one run of it...

Thanks to HopRope for pointing out that the J version actually has a second quest at all :)