Chilis are from the New World, and as CMosychuk [has answered][1], belong in the *Capsicum* genus. Their name [derives from the Aztec Nahuatl][2] word for the plant `chilli`, and this name has been used by the Aztecs to refer to the plant prior to Spanish contact and spread of the plant to the Old World. 

The pepper, however, belongs to the completely unrelated *Piper* genus, and the plant originated in the Old World. Its etymology derives from the [Sanskrit word][3] `pippali`, also referring to the plant. As the Etymonline entry suggests, the word "pepper" was also used to refer to the *Capsicum* plants when they were first introduced to the Old World, perhaps due to their similar spicy taste. 

Therefore, it is not the case that chilis are peppers, since the *Capsicum* genus has no recent genetic relation to the *Piper* genus. 


  [1]: https://biology.stackexchange.com/a/43370/10456
  [2]: http://www.etymonline.com/index.php?allowed_in_frame=0&search=chili
  [3]: http://www.etymonline.com/index.php?allowed_in_frame=0&search=pepper