The nature is exactly what they're talking about. This is what the researchers actually said:
"We observed that the heart was acting as though it had a mind of its own and was profoundly influencing the way we perceive and respond to the world. In essence, it appeared that the heart was affecting intelligence and awareness."
Sounds as though they're also talking about consciousness when they mention "awareness".
Here's the link, again:
http://www.heartmath.org/research/science-of-the-heart.html
They seem to be categorically suggesting these neurons are of the type found in the brain. Here are the actual slides for anyone curious:
http://www.heartmath.org/research/research-our-heart-brain.html
It may not be just a matter of a new personality arriving with the new brain... According to this research, the host heart may have some input, too. It might be that we get an entirely new person, a strange sort of hybrid, and perhaps very different from both former individuals. A bit like the symbiont lifeform in DS9...