In terms of standard notation, the pattern of 5th relations is not really a circle but a spiral, and the points where one arc of the spiral parallels another are generally not interchangeable, but only correspond in pitch due to equal temperament. The extent of the spiral is from Fbb to Bx, though for common practical use one can generally get by with only the range spanned by single accidentals (Fb to B#). For most applications of the 5ths series, I agree with Jack--a "ladder" or set of strips shiftable in the manner of a slide rule comes in handier than a circle or even a set of nested wheels.
Consider the following set of strips:
Strip 1: The Base
Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
E B F# C# G# D# A# E# B# Fb Cb Gb Db Ab Eb Bb F C
Strip 2: Transposition Map
Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
Strip 3: Scale/chord degrees
V II VI III VII IV I V II VI III VII IV
Strip 4: Modes
MAJ MIN
|Lyd Ion Mix Dor Aeo Phr Loc|
Maj Maj Maj min min min dim
The second line of notes on the Base strip are enharmonic equivalents. To avoid confusion, those notes should ideally be printed in smaller type and enclosed in parentheses, since they will be referred to relatively infrequently. This is one of the great failings of the wheel--that one too often sees enharmonic equivalents that are the improper choice for the desired context. The transposition map is merely a duplication of the top line of the Base, and in fact may be preferable to use in place of the Base when enharmonics are irrelevant. (Actually, orienting these strips vertically, in "ladders" and Jack suggests, is probably easier to read, and longer names can be used. But I'm limited by the medium of text.)
First, let's look at the Base. It's the sequence FCGDAEB repeated three times, first with flats, then naturals, then sharps. (Memorize this sequence forwards and backwards, since it pops up continually.) The enharmonics row just repeats the opposite end of the previous row, as if the ends were overlapped in a circle (as on a more complete spiral of fifths).
Now, place the Modes strip so that "MAJ" is positioned over a selected major key on the Base, like A, and place the Degrees strip between them with the "I" underneath "MAJ" as well:
MAJ MIN
|Lyd Ion Mix Dor Aeo Phr Loc|
Maj Maj Maj min min min dim
V II VI III VII IV I V II VI III VII IV
Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
What does this tell us? First, the notes within the A major scale are those that fall between the vertical bars: D A E B F# C# G#; so we know the key signature is 3 sharps (and in fact, they're notated on the staff in the order given; flat signatures appear in reverse order: Bb, Eb, Ab...). Using a typical circle of fifths, you'd see Db and Ab instead of C# and G#; now maybe what Jack is saying becomes a bit clearer. Yes, in equal temperament, they have the same pitch, but in proper notation, they're entirely different beasts.
We find that the relative modes for A major (or Ionian)--those which share the same key signature and note set--are D Lydian, E Mixolydian, B Dorian, F# minor (or Aeolian), C# Phrygian, and G# Locrian. What about enharmonics like Db Phrygian? Ain't no such thing, because (checking the enharmonics row) there are no single-accidental equivalents for two notes in the scale--D and A--and standard key signatures don't use double-accidentals (Ebb and Bbb would be required). Is that readily apparent from the circle? Nope.
We find that the common chords of A major, per the third (Maj/min/dim) row, are: tonic (I) AMaj, subdominant (IV) DMaj, dominant (V) EMaj, supertonic (II) Bmin,
submediant (VI) F#min, mediant (III) C#min and subtonic (VII) G#dim. Note the degree order: it just interleaves I, II and III between IV, V, VI and VII. The primary chords for any major key can be read directly off the Base strip by itself: IV is to the left of the tonic, V to the right, and all three are major.
Now, let's shift our point of reference to F# minor, by the simple expedient of shifting the Degrees slip so the I is under "MIN". The same notes and common chords are used, but their interpretation relative to the tonic is now different: I, IV and V are now all minor chords (F#m, Bm and C#m), II is diminished (G#dim) and III, VI and VII are major (A, D and E). In fact, VII becomes a primary chord in minor mode.
The most common modulations are up/down a fourth/fifth. Up a fifth is down a fourth, and vice versa. On the ladder, this means shifting the tonic to a neighboring rung. The next most common modulations are up/down a third, typically between relative major and minor keys--we've already dealt with that, but as josepp described, major to minor is a shift clockwise (left) three steps and minor to major is a shift counterclockwise (right) three steps.
Now we come to one of the best advantages of the ladder over the circle: transposition. Slip the Transposition slip over the Base slip so the Base indicates the source key and the Transposition slip the destination key. Now you can read the proper transpositions directly, with no confusion regarding enharmonic equivalents. It may help to position the Modes slip over both, so that the range of notes in the scales, per the mode, is more apparent. Try it with keys not close together (say, transposing by a mere semitone) and compare the results with what you'd read from a standard circle of fifths.
Jack is right: even though the equal tempered scale may be circular in terms of pitches and enharmonic equivalents, much music is not equal tempered, and most tonal music is governed by theory and notational practice that does not treat enharmonic pitches as interchangeable. The harmonic applications of sequences of fifths never encompass the entire circle but rather, as Jack asserted, only operate within a subrange of what is better described as a linear continuum than a cycle. This is where the ladder shows its superiority over the circle, despite the ubiquity of the latter in common dedagogy. I believe the popularity of the circle lies more in its sexiness of presentation than in its practical utility.