A phlebologist is a doctor who specialises in the treatment of diseased veins. The diseases that they treat include, most commonly, varicose veins and spider veins. Other problems include deep venous thrombosis (DVT), venous ulcers and all the issues around chronic venous insufficiency (scarring around the ankles, pigmentation, staining, ulcers etc). There are other problems which phlebologists can help with including leg swelling, lymphoedema and unusual deposits of fat in the lower limb called lipoedema.
Vascular surgeons treat vein problems as well, but also manage disease of the arteries. It does not matter who treats your veins problem as long as they are trained properly and have an interest in the management of venous disease.