There's a fairly intuitive reason for this--"intuitive" in this case meaning that I'm not going to give a formal music-theoretic reason, but rather try to simply provide some intuition about why this might be so based on the circle of fifths and how "close" and "far" particular tonalities are from each other.
First, some observations:
- Major and minor pentatonic scales are the "nicest" notes that can be grouped together (to Western ears). This is a fuzzy concept, of course, but it's easily formalized: notes (and keys/chords/scales) that are closer together on the circle of fifths sound more similar and are "nicer" together than notes that are further apart on the circle, and any five consecutive notes on the circle form a pentatonic scale.
- The major scale is a pentatonic plus the two "next-nicest" notes--i.e., any seven consecutive notes on the circle of fifths form a major scale. (Note that in order to take a pentatonic scale and form a major scale of the same key, you must add the two notes directly before and after the pentatonic scale group on the circle of fifths. More about this issue of "directionality" below.)
- Notes (/keys/chords/scales) that are a tritone apart are as dissimilar as possible; they are diametrically opposed (i.e. maximally far apart) on the circle of fifths. This typically means they sound the least "nice" together (hence the term "Devil's interval").
- This concept of "similarity" and "dissimilarity" is transitive. For example, C neighbors G on the circle of fifths, and G neighbors D, so C and D are fairly similar. On the other end of the spectrum, C and F♯ are a tritone apart and are thus maximally dissimilar; F♯ is next to C♯ on the circle of fifths, and so C and C♯ also clash (though not as badly as F♯ and C).
Now, sticking with our example of using C as the basis for our analysis (but, as mathematicians say, "without loss of generality"), we can see that:
- Notes in the C major pentatonic scale are "similar to" C (i.e. they are the next 4 notes going up the circle of fifths).
- Notes in the F♯ major pentatonic scale are "similar to" F♯, and F♯ is maximally "dissimilar from" C, so, transitively, we've now divided up the keyboard between "notes that are similar to C" and "notes that are similar to F♯ and maximally dissimilar from C."
- We've got two notes left over: B and F (E♯). These belong to both the C major scale and the F♯ major scale. Thus the property you noticed is (of course) reflexive: if we're in F♯ major, the unused notes are the C pentatonic scale, and if we're in C major, the unused notes are in the F♯ pentatonic scale.
This leads to an interesting thought: we can evenly partition the keyboard into two 6-note "major-ish" scales. Presumably we'd want to give B to C major and F (E♯) to F♯ major to ensure that we have leading tones for both scales.
The biggest "fuzzy" element of this analysis is that "similar" (as used above) doesn't precisely correlate with "nearness" on the circle of fifths, because going backward from C (i.e. moving up by 4ths or down by 5ths) quickly leads to B♭ (A♯), which is part of F♯ major pentatonic. But there's no particular reason why "closeness" on the circle of fifths shouldn't be considered reflexive (i.e. there's no reason to consider D closer to C on the circle than B♭ is). This part of the analysis could be made more rigorous by introducing a concept of directionality into the concept of "similarity" with which we've built the scales in question, i.e., by formalizing a reason for going up by 5ths when constructing our scales rather than going up by 4ths. This is well understood by applying the concept of the tonic-dominant relationship, but that's beyond the scope of this answer.
Can we use this when playing music? Matt L. has already given one use-case, unsurprisingly coming from the world of jazz where the concepts of "nearness" and "farness" are bent pretty far from how they're used in Western classical music.
But even in more "traditional"-sounding music, and especially in pop songs and Broadway-style show tunes, there's another use-case: dramatic modulations. If you're in C major and you want to have a sudden radical shift that still sounds "right", why not modulate as far away from C as possible? If you start playing the F♯ pentatonic scale, suddenly, most of the notes you're using would have been (somewhat) out of place in C major, but in their new context post-modulation, they're as close as possible to your new tonic (F♯). The shift is so dramatic, and the pentatonic scale is so harmonically "clear" or "obvious" sounding, that composers will often simply modulate without any real transition chords; you can jump right from C Major to F♯ Major without preparing the listener, and the chordal structure will still remain clear.