What is the main core difference of The Flash, that no other speedsters possesses?

It's... complicated. Due to the Flash's extensive history in comics, cartoons, tv, and soon to be movie, there is no one definitive list of what he can do. And there are multiple Flashes, Jay Garrick, Barry Allen, Wally West, etc. The CW Flash is not a straight forward copy of comic Barry Allen, but a mixture of them.

That said, current comic canon (not yet rewritten in the New 52) is that Barry Allen is the source of the Speed Force which every other speedster uses. In essence, this makes him more powerful than all the others, and he could do anything they can, if he learns how. So there is no upper limit on what he can do compared to others.

In The Flash: Rebirth #4, Max Mercury revealed to Barry Allen that the Speed Force was created by Allen, himself, when he became the Silver Age Flash; adding that, "When [Allen runs, he generates] the kinetic wall between the present and the time barrier. It's an electrical energy that exists in every dimension, every universe, and every era. It touches every part of reality. It contains the knowledge of every place and time. The Speed Force is the all-encompassing Flash Fact."

Reverse-Flash, Eobard (in the comics known as Professor Zoom at times) is essentially using Barry's old costume, in the future. His power levels depend on Barry's power levels, but in the future. This is why he need to make Barry Stronger, in order to retro actively make himself stronger. Yay Temporal Mechanics. When he changed the past and almost killed Barry, he changed the amount of Speed Force he could absorb.

Edit: In S02E10 "The Reverse-Flash Returns", a younger Eobard says that he replicated Barry's formula. This implies a version of "Velocity 9" or "Speed Juice" in Earth 1's future. Or it's an exact copy of Barry's accident. In either case, it would mean Eobard didn't get Barry's old costume, and makes most of Season 1's wheelchair and tachyon plots "wrong"? More details required.

This is essentially also what Zoom, who resembles the Black Flash (A Grim Reaper type character), also wants. To fatten Barry up like a turkey to take his power. To drain him of the Speed Force.

There is no single power that they want of his, they want the Speed Force in his cells. The way the Speed Force powers work, they can all copy one another with practice. Most of the Flashes can travel through time. Most of them can vibrate through things. Wind Vortexes. Speed Mirages. Super Punches. This is the trick to the Speed Force, understanding it and applying it in creative ways.

_{Side note: All the Flash enemies tend to use the same names at various times. Eobard has gone by Professor Zoom, Reverse-Flash, Black Flash. So did Zoom, who in the comics is Hunter Zoloman, Wally West's reverse-flash. It's confusing.}

Reverse-flash can time travel too, he did it in episode #10, I guess but really didn't get why he hates the flash. They said zoom wants flash's speed to become stronger and faster and because he doesn't accept the fact that there is another speedster but him on any earth. He wants to be the only speedster so he took down Jay Garrick on his earth and now it's Berry's turn, they clarified only that much for now.