While I think Troff's answer is very insightful about the environment inside Atari at the time, I think it might have something more to do with how licensing of popular arcade games was handled at the time.
In particular, it was common practice for a company such as Nintendo to sell the rights for a game based on the target platform. For example, Nintendo licensed Donkey Kong to Coleco for the home game consoles while Atari received the rights for the home computers.
While it would be wonderful to simply cap the answer at who had the rights, it seems that the playfield is riddled with exceptions. For example, when the Coleco ADAM version of Donkey Kong was released (considered a home computer), the version on that platform was developed by Coleco, not Atari as you would have expected from above. Coleco seemingly felt that since the ADAM was an expansion module of the underlying ColecoVision game console, the ADAM port was just an enhanced ColecoVision version and thus within the terms of their license agreement. This ended up as a lawsuit raised by Atari and ultimately caused the ADAM version to get pulled, but not before it had already been widely distributed.
In regards to Mario Bros., it does seem that the 400/800 and 7800 versions were developed by Atari Corp. while the 2600/5200 (*1) versions were developed by Atari, Inc. Historically, Atari Inc. was the original Atari company that Nolan Bushnell founded while Atari Corp. was the Warner sell-off of the home computer and console division to Jack Tramiel in July 1984.
Given the timing of the publishers, it would suggest that perhaps Atari didn't have the rights to the home computer versions when the original 5200 port was developed and when they finally did acquire them, they rewrote the game; perhaps the original 5200 code wasn't suitable as a basis for porting to the other home computers? I haven't done a comparison of the game across platforms, but perhaps there is some commonality across the home computer versions.
*1 - The 7800 wasn't released until after Tramiel took over.