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Straight from Wikipedia:

Thrash metal is a subgenre of heavy metal music, one of the extreme metal subgenres that is characterized by high speed riffing and aggression. Thrash metal songs typically use fast, percussive and low-register guitar riffs, overlaid with shredding-style lead work. Thrash song lyrics express nihilistic views or deal with social issues using visceral, gory language, an approach which partially overlaps with the hardcore genre.

The origins of thrash metal are generally traced to the late 1970s and early 1980s, when a number of bands began incorporating the sound of the New Wave of British Heavy Metal (NWOBHM), creating a new genre and developing into a separate movement from punk rock and hardcore. This genre is much more aggressive compared to its relative, speed metal. There is often significant crossover from one metal category to another, and some bands use musical influences from non-metal genres, including classical music and jazz.

Power metal is a style of heavy metal music typically with the aim of evoking an "epic" feel, combining characteristics of traditional metal with thrash metal or speed metal, often within symphonic context. The term refers to two different but related styles: the first pioneered and largely practiced in North America with a harder sound similar to speed metal, and a later more widespread and popular style based in Europe (Especially Germany, Italy and Scandinavia) with a lighter, more melodic sound and making frequent use of keyboards. With mainstream metal's fall from popularity and the growth of the underground metal scene, power metal can be seen as a parallel counterpart to extreme metal within the metalhead subculture.