Basically these are synonymous terms: "leads to" and "resolves."
A very important concept in tonal harmony is the strong sense of movement in the half steps of the diatonic scales. In solfege these are the tone pairs
Things can get confusing if we discuss the leading tone and resolution_and_ jump between different musical styles. You mentioned cantus firmus which suggests 16th century counterpoint, modal scales, and the species counterpoint teaching of Fux. In the 18th century homophonic style the leading tone and harmony have a different treatment with parts rooted in the older style. Let's look at the two separately. Just remember: two different styles, two different approaches.
The quote "...penultimate note should go to the final in a stepwise manner..." is describing the clausula vera, the final cadence in 16th century style. The idea is the final tone - the tonic - should be approached by two voices in contrary motion one voice moving by a whole step and the other moving by a half step. When the final is approached from below by a half step that penultimate tone is the leading tone. The exact handling depends on the mode so let's just run through each mode.
In Ionian and Lydian modes the tones above and below the final are a whole step and a half step. Let's use
C as a final. The whole step above is
D and the half step below is
D move to
C to end this is the clausula vera. The
B a half step below the
C is the leading tone.
In Dorian, Aeolian, and Mixolydian modes the tones above and below the final are both whole steps. Let's use
D as the final. The two whole steps are the
E above and the
C below. If we use those diatonic pitches and move
D to end it is not a clausula vera! One voice needs to be altered to make a movement by half step. The convention is to raise the tone below the final by a half step. This will make our
C become a
C#. This gives us the correct movement for a clausula vera and the
C# has become a leading tone (because it is one half step under the final.) Note: The unaltered tone a whole step below the final can be called the subtonic.
In Phrygian mode we have a unique situation. The tones above and below are a half step and a whole step. Instead of the half step being located under the final, it is located above the final. Technically we fulfill the definition of a clausula vera, because when the voices move to the final one moves by whole step and the other by half step. But we don't really have a proper leading tone below the tonic. The tone above is sort of like a leading tone. This half step approach from above give a unique flavor to the Phrygian mode the other modes don't have. To illustrate, let's use
E as the final. The half step above is
F and the whole step below is
F move to
E we have a clausula vera without altering any tones with accidentals, but technically there is not a leading tone.
Briefly, let's consider 18th century harmony. During that time period the modes were no longer used. Instead harmony used the major/minor system. In that system we can shift the tonic around temporarily using tonicizations and modulations. Without going into too much detail, any diatonic major or minor chord in a given key can temporarily become the tonic. The root of such chords will become the temporary tonic, and the half step below that tone will be the temporary leading tone. For example, in
C major the chords
G major, and
A minor can all become temporary tonics. The tones a half step below each of those roots will be temporary leading tones -
You can link the 18th century style back to the 16th century style by constructing a clausula vera to lead to any of those tonics. But in modern terminology we don't call it a clausula vera. Instead it is called a secondary or applied dominant and label them with Roman numeral analysis symbols. For example, the temporary leading tone move to
D in the key
C major would be labelled like:
I packed a lot of theory into a very short description. Take time to digest it and continue to read up on these topics.
- the leading tone is a half step below a final/tonic
- half step movements in the diatonic scale have a strong sense of pull or movement
- resolution and important harmonic movements often involve half step movements