Nope, completely different. Punk is speaking for longtime wrestling purists. Austin was the all around anti-authority figure that you didn't have to be a longtime wrestling fan to relate to. Austin brought people into wrestling fandom. You didn't have to be an expert on pro wrestling to know what Austin was about. You have to already be a longtime wrestling fan for Punk's promos to mean anything to you.
Punk is more Shawn Michaels in 1996, to the point I think the current storyline is directly inspired by HBK's one then, just with a modern twitter era spin. Shawn started doing these "shoot" promos about how frustrated he was with the company, saying things you just didn't say back then, breaking kayfabe, talking about older stars (like Bret Hart) not wanting to allow younger stars to rise, calling Sid "the WWF's most expensive piece of luggage," that they were all carrying around the country, etc. It re-invented Shawn's character and led to DX.