Unfortunately, Muppets Tonight was added to ABC's lineup at a time when the station was struggling and trying to find a strong identity...and in the process went through several management changes - each with their own ideas as to what they wanted the network to be. It was not uncommon for shows to debut with much fanfare and then to exit noiselessly not long after. Muppets Tonight fell in this category.
When they were initially signed, the network brass was very supportive. However when it came time to air, the programmers started doing the things that they didn't want to happen when they came up with the idea - they promoted it more as a kid show and initially placed it in their Friday lineup geared towards tweenie sitcoms. Even though the show debuted with a publicity blitz, follow up episodes were barely promoted (it's not uncommon to meet Muppet fans who go "there was a Muppet Show on ABC in 1996?" Really?"). Despite this, ABC in its characteristic schizophrenia of the time , renewed the show for a second season and proceeded to look for a new time slot for the show (putting it on Sundays paired with America's Funniest Home Videos). but even then ordered only nine episodes for the enxt season instead of the original's thirteen. Finally the decision was made to not air the remaining episodes on ABC but rather on Disney Channel (both stations being under the same corporate umbrella).
Disney at least seemed happy to have the show on its channel and aired all 22 episodes. But being on a cable channel and carrying a for-kids stigma probably didn't help.
At least, Fox, though not without their faults - is usually committed to their shows and tries to give each of its programs a hip spin so if nothing else, the partnership with Fox should enable them to shake some of the public's Muppet=Kids Entertainment mindset.
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As to why The original Muppet Show ended, that was purely by Henson's choice. The show was syndicated and had a bit more freedom as to its longevity. All involved felt that after five years, they had reached their creative peak (originally they never even thought the series would last more than three years) and wanted to leave it while it was still at the top of its form instead of running into the ground. They also wanted to move on to more varied projects like specials and movies.
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