-The packaging and special features are different. All 5 of Rhino's sets come with animation cells, only season 1 of the Shout factory ones comes with magnets (or the matrix box set).
-Cost. Rhino's sets may be cheaper now, but when they first came out they retailed for about $59.99-$69.99 in Canada. The shout ones gor for about $19.99 - $29.99.
-Rhino remastered the video from the pre broadcast versions of he show because the masters (the ones that aired) were toast. This meant that there were way more errors in the video that were corrected for the broadcast version originally, which had to be corrected again. Personally I think Rhino did a pretty good job, but there are hard cores who insist that they failed.
Shout! basically did the same thing, but fixed the mistakes Rhino forgot or never noticed (still forgetting some themselves) and went the extra mile and reintegrated some of the master footage into some segments of the episodes. Now there are some errors that made it into the master tapes and Shout! decided it was best to not correct them and to leave them as close to as they were originally aired as possible.
-Rhino remastered the audio and added in some extra sound effects. Shout! just remastered the audio and did not add anything in.
I only watched through the Rhino set once and for the most part, it was exactly how I remembered it as a kid. I also bought the new Shout! Matrix boxset and have gone through about 1/4 of it. In all honesty, I didn't notice any of the differences until someone pointed it out and I didn't really care either. Both look really great, but in the end I do think the Shout! sets are superior.
Info about Rhino releases:
http://tfwiki.net/wiki/Kid_Rhino
Info about Shout releases:
http://tfwiki.net/wiki/Shout!_Factory
