A blank flame has only a linear or linear3D variation and pre/post transforms that don't touch the point. So it starts with a random point and iteration doesn't change the point, so that's what you see. Apophysis restarts with a new random point every 10,000 iterations, resulting in lots of points in the flame. JWildfire does not, so there is just one point for each core. If you have a four core system, you see four blobs; if you have a 32 core system, you see 32 blobs.
In JWildfire, they are blobs instead of points because they are anti-aliased. If you go to the Anti-Aliasing/Filter tab and set Antialiasing amount to 0, you'll see four points instead of four blobs.
If you move the transform slightly, each iteration will shift the point slightly. If the move is very small (try 0.0001), the resulting points are close enough together to show as lines. Otherwise, they are just too spread out to see. Even better, try 0.00001 with progressive render enabled; you can see the points march across the screen!