'Pretense' can mean a claim with no facts to support it or false claim, a lie. It can also generally refer to false behavior. "The organization supported the new drilling on the pretense that its effect on the environment would be minimal." (not supported) 'Pretext' is more specific and refers to a false claim, a lie. "He left on the pretext of a family emergency." (a lie - you could have used 'pretense' here, though)