According to western notational convention, F in the first case will resolve downwards and B upwards. E# on the other hand signifies a change in functional status and will resolve upwards to F# as a leading-note would. In both cases, the functional role of the lower note affects the movement of the upper note. From a listener's point of view, we cannot then be sure how this tritone, if heard in isolation, will proceed. Or, to put it differently, the composer can resolve in either way and it will sound logical to the listener. Listen for yourself.
Once you understand this phenomenon, you can easily appreciate how composers have exploited this enharmonic equivelence to create interesting harmonic "twist". Two very common chords that undergo enharmonic reinterpreations are
(i) the augmented sixth chord (usually the Ger 6/5), and
(ii) the diminished seventh chord.
Ex. 26.14 is an instance of a V7 chord switching to function as a Ger 6/5 chord, thereby shifting the key downwards by a semitone. This is an extension of the idea of pivot chord--one that involved enharmonic re-spelling (but aurally, it really sounds the same). Funcationally, in this case, the chord changes from a dominant function chord to a predominant one.
If we reverse the enharmonic change, that is, re-spell a Ger 6/5 chord as a V7, the modulation will be a semitone upwards. This is a common trick some 19C composers play to create remote, hence surprising, modulations. Taking this one step further, the V7 chord can be an applied dominant. Can you now understand why Ex 26.10(a), unlike (b) and (c), does not modulate a semitone away? See Ex. 26.11 for a musical example of what is sometimes referred to third relation. This latter term distinguishes itself from the diatonic form of third relation, namely the minor-third relation between relative keys (e.g. Ab maj - F min) and the mediant key relation (e.g. Ab maj - C min). We may of course also think of the C major in the context of Ab major as a mixture-third relation, which is essentially what the special term "third relation" is about.
Enharmonic re-interpretation of the diminished seventh chord works in much the same way. Here, each of the four chord tones can be enharmonically respelt and the voice-leading tendencies will change accordingly. Ex. 26.15 illustrates this very clearly. Can you now examine Exx. 26.16 & 26.17? Be sure to listen to them to hear the harmonic surprise.
We are familiar with enharmonic equivalences. For example, C# can be spelt as Db depending on the context, and augmented fourth sounds like diminished fifth on the equal-tempered scale. In voice-leading terms, however, there is a difference: