What is difference between an artery and a vein in terms of their elastic tissue?

Arteries are made of 3 distinct layers. A intimia, media and adventitia layer. The media layer is the one that contains all the muscular fibres as well as elastin.

In elastic arteries the media is made from a lot of elastin and not much smooth muscle. These are generally found near the heart (such as ascending aorta) because they need to be able to expand with the large pressure being pumped from the LV. Muscular arteries however have a lot of smooth muscle and not much elastin. These are found away from the heart in areas such as limbs.