Hi Psyche. The Coolglide is an appropriate laser for small spider veins. The long pulsed Nd:Yg is a very common type of laser to use for spider veins on the face and extremities. With that said, the outcome of your treatment does not appear normal. If you are healed from this procedure now, but have some residual scarring then you should approach your practitioner about taking some remedial action. If you do not want to return to the practice, then you should consider asking them to pay for the procedures you may consider.
Fractional laser resurfacing may be a good option at this point to help with the scarring. Good luck.