There is no cure. I completely agree with dr. Rosen. It is important to remember, though, treatment makes veins go away. However, there is no guarantee that you won't get more veins in the future. Varicose veins and spider veins occur primarily because of genetics and no one can change that. That means that you have a tendency toward developing bad veins and you will always have that tendency.
Sclerotherapy. Sclerotherapy using detergents such as polidocanol, sotradecol, morrhuate; glycerine with xylocaine, (lidocaine) and injected using a small bore needle is best treatment method. This will eliminate sv, but is a process requiring several treatments, depending upon severity. Sv can come back in other sites, and generally touching up every year or two may be necessary.
It depends... Treatment choice depends on size and location of spider veins. Sclerotherapy is generally advised for veins big enough to enter with a needle. Surface laser for those too small to inject. Usually results are permanent. Recalcitrance may suggest a deeper source of reflux nearby that needs to be dealt with. See a board-certified phlebologist to assess your needs and tailor your care accordingly.