In time, do the same veins return after sclerotherapy, or do the treated veins remain permanently destroyed?

Most likely after sclerotherapy new veins present themselves. It is possible that a vein that was previously treated and did not go away can get larger if still present, but classically new veins develop.

In general, if Sclerotherapy erased a vein or group of veins for any time period, then those particular veins are destroyed. However, with age and gravity, new veins can develop in the same site, or a "sunburst" pattern of veins can occur all around the treated site. If veins recur after years, re-treating with Sclerotherapy may be appropriate, but if they recur in under a year, consult your Cosmetic Dermatologist for their best advice.

Sclerotherapy is a very successful procedure when done by an experienced injector.  Despite this, some spider veins will reform.  Usually these are new veins that will grow over time.  Spider veins are due to genetics, aging, trauma, hormonal changes and sometimes unexplained.  When considering spider vein sclerotherapy, remember that some maintenance is required over periodic intervals.  Also, if you have extensive spider veins you should have a venous reflux exam to see the status of your saphenous systems.

You will develop new venous disease if you don't take preventative measures - exercize legs, keep calf muscles strong, wear compression stockings, take bioflavonoids or horse chestnut extract. The sclerosed veins should be successfully sclerosed if properly injected. 

If a vein is injected properly with sclerosing injections, that vein will permanently go away. If you are keen to get these types of veins you can develop new ones over time, but if you get a maintenance treatment every now and then you should never have the same amount as you did when you started the treatments. "This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."