Symthic provided the data files for Battlefield 4 (but the actual link to the data appears to be dead).

I don't think you can

I think it is not possible to get RAW gamedata of Battlefield 4. You either would have to break the law¹, or ask DICE (LA) for the original data.

Checking out the game files yourself

I have looked in the Battlefield 4 files that I have on my pc. I did find some files that I suspect store the data in them, but like almost all the game files, I couldn't open nor read these files. These are most likely datafiles that were generated when the source code were build to the actual program that runs on your computer.

For example, I found

\Battlefield 4\Data\Win32\XP0Weapons.sb

and

\Battlefield 4\Update\Xpack3\Data\Win32\XP3Vehicles.toc

I suspect that these are files that are read by the .EXE application when the game is started. But outside of that, there would be no other programs that can read them.

So if you would like to retrieve information directly out of Battlefield 4's yourself, you would have to reverse engineer these files. This however is impossible without violating Battlefield's 4 EULA¹:

Except to the extent permitted under applicable law, you may not decompile, disassemble, or reverse engineer the Software, or any component thereof, by any means whatsoever.

So how does Symthic does it?

I'm really not sure. My guess is that either they have received some special license that allows them to reverse engineer the software, or they received the RAW game files, the original source code and data DICE (LA) created, before the game was build.

I have searched Symthic's site and forums for more information regarding their source, but I was not able to find any, except their claim that "BF4 info is taken directly from the game files". Outside of Symthic I could also not find any solid proof for accuracy and trustworthiness of Symthic's data.

Footnotes

^{1. I am not a lawyer, this is not legal advice.}

This is game dependent, some games share a trophy list across PS4 and PS Vita, so unlocking a trophy one one device will also unlock it on another. This is particularly true for games that are "cross-buy".

Other games have separate trophy lists, which means unlocking on one platform won't affect the other. Minecraft is an example of this, where each version has a independent list. And even though you can transfer the save data between the games, it won't retrospectively unlock the trophies you've previously unlocked.