Is Electronic Band Gap ( EBG) and Metamaterial same?
So in recent years, metamaterials have become a hot topic for the antennas and propagation community. Although some would argue that there are formal definitions, what most people call "metamaterials" today are structures which exhibit non-traditional behavior not commonly found in nature.

In the early 2000s, Pendry and his group published a paper talking about building a "cloak" from negative index of refraction "metamaterials." This spawned a huge interest back into this field, and over the next several years, prominent scholars in applied electromagnetics presented and debated all sorts of cool things (right-handed, left-handed, causality, etc). A good source on the subject is the book Metamaterials: Physics and Engineering Explorations by Nader Enghetta and Richard Ziolkowski.

Artificial dielectrics and impedance surfaces were around for years. The called them many things in papers, but they are all structures which exhibit strange scattering behavior. There's a chapter in Enghetta text on "impedance surfaces," which is the same thing as EBGs; Daniel Sievenpiper wrote it. Basically, they're structures which appear as an artificial impedance boundary to incident waves. Sievenpiper and his group at UCSD pioneered much of the research in this area. They came up with the now famous mushroom periodic structure. An application of this is building artificial magnetic ground planes, which if possible, would be huge. If you're familiar with image theory, think about what happens to horizontal electric currents close to a magnetic ground? (It doubles!)

Anyways, I don't want to open the floodgates of debate on what constitutes one or the other. In general, you'll find that most people today consider EBGs to be a type of metamaterial, along with frequency selective surfaces (FSS), magnetodielectrics, metaferrites, negative index of refraction materials, etc. But do check out the Enghetta/Ziolkowski text.