I have vias connected to the polygon. In the old design (left pic) the vias have annular rings. But I modified few other things in the PCB. I removed unused vias in both new and old design. But now in the new one (right picture) the vias don't have the rings showing up. I guess, it doesn't make difference for the manufacturer? But also, I don't know why is it happening.
If you check off “tented” in the via properties on the relevant side the solder mask will not be pulled back around the via, so it will look different in both the layout and when it is fabricated.
See, for example, this answer.
There are various reasons to prefer one over the other, but that’s outside the scope of this answer.