In some time signatures, they’re not the same thing.
Take moderate or fast 6/8 for instance. There’s six counts — six eighth notes in each measure — but only two real beats on count one and count four. This is the nature of 6/8, that lilting feel of three in each beat, but only two beats in each measure.
Cut time is like this as well. There’s only two beats in each measure, but we know that a measure of cut time contains the same number of quarter notes as 4/4. Cut time is a feel more than anything.
Some pieces feel better when the beat encompasses more than one count. Take any Gigue, for example. A Gigue feels more correct (and capture the right mood) when each beat encompasses three counts. Here I’m generalizing, most gigues are in a triple meter like 3/8 or 3/4. The first movement of Leo Brouwer’s El Decameron Negro is in 5/8, but doesn’t feel right with five beats in each measure; it feels right when there’s one beat that encompasses all five counts.
What the time signature really means is a question you should ask yourself when trying to figure out the overall feel.