What you are seeing is actually not a preference towards sliding the feet or not sliding the feet. It's actually a tiny shift in where the fighter's balance is.
When a person stands they will usually have their weight over the heel of the foot. When they walk they will put the heel down first. It's the natural way of walking. This is the way they are standing and walking in the Muay Thai video.
However many martial arts will advocate keeping the weight over the ball of the foot (that is to say close to the toes) instead. This is what you are seeing in the 'MMA' style videos. I believe this is more common in styles with a Japanese influence. In the style I learn (Japanese Shorinji Kempo) for example we always use this way of standing once we are advanced enough to have learned it.
As for the advantages or disadvantages...
The traditional Muay Thai method allows for very heavy kicks however kicks of this kind can compromise balance and require you to change stance because you are kicking through the target.
The weight on the ball of the foot is favoured if you prefer a light kick that doesn't follow through. You might call this a 'snap kick'. This is particularly useful if you are point fighting rather than full contact sparing because although your kick won't have as much power it will be faster and more controlled.
The end of this video by Jesse Enkamp shows the difference in kick style.
Muay Thai from what I have seen has a large aspect involving catching legs as well. Having your leg trapped is probably less hazadous if your weight is further back.
In conclusion look at the overall stance rather than the contact with the floor. The stance drives contact or lack thereof not the other way around.