Why, for example, does the word "hello" sound completely different to the word "goodbye", or the letter "a" from the letter "b"?
I know it can't be pitch, because all of these words and syllables can be spoken at the same pitch and still sound distinct, and changing the lyrics of a song does not change the pitch.
What musical property is it then that makes words sound different from each other?
As Michael Curtis has pointed out, from the linguistic side, the study of phonetics is all about what speech sounds humans make and how they make them. Phonetics doesn't really approach things from a musical perspective, so I thought I might try to make some correlations between phonetics and musical acoustics.
Phonetics divides speech sounds (phonemes) into two broad categories: vowels and consonants. The lines can be a bit blurry there, but vowel sounds always involve the vocal cords and usually made with the mouth more or less open, while consonants involve specific motions of the teeth, lips, and tongue and may or may not use the vocal cords.
For vowels, we always use our vocal cords, which means vowels always have some pitch. The pitches used during speech generally do not have a typical musical relationship, but sometimes might be "accidentally" musical. For instance, when a child taunts on the playground something like, "Johnny is a chick-en!", they often us a sing-song tone that is a melodic minor third. But that's incidental.
The way we make different vowel sounds is by changing the shape of our mouths, and this changes the timbre of the sound made by our vocal cords. Another musical way to look at it is that we are filtering (like with EQ or a synth filter) the pitch that is created by our vocal cords.
That entirely covers the musical aspects of vowel sounds. We could talk about loudness and duration (the two other main dimensions of music), but neither of those change the vowel sound we make or hear.
Consonants are more complicated. Let's divide them into the phonetic categories of voiced (using the vocal cords) and unvoiced (not using the vocal cords).
Unvoiced consonants (like /t/, /p/, /f/, /k/, /s/), from a musical standpoint, are closest to percussion sounds. These are the kinds of sounds we make when we beat box. Percussion sounds and unvoiced consonants are both musically unpitched, and instead distinguished solely by timbre. The two main timbral elements of these sounds are the envelope and formant. The formant is like a filter setting, just like for vowel sounds, but since there is no pitch to filter, what is being filtered instead is noise or unpitched tones. Unpitched tones are groups of frequencies that do not have a harmonic relationship to each other, so we don't hear them as a note. Think of two different cymbals, a "high" one and a "low" one as being examples of noise with two different formants.
For unvoiced consonants, there are two subcategories we can talk about, plosives, fricatives. Plosives (/t/, /p/, /k/) have a very short loudness envelope that reaches maximum volume very quickly and then dies away just as quickly. This is most similar to a drum sound. The different sounds of plosives come from their different formants. In this case, it's mainly how much and what kinds of noise is being made along with the plosive sound. A /p/ sound has essentially no noise, like a kick drum, while /t/ and /k/ have two different kinds of noise that are more like a hi hat and snare drum, respectively. Another thing that makes the /t/ sound different from the /k/ is the position of the mouth is different, which causes different filtering just like we see in the vowel sounds.
Fricatives (/f/, /s/, /sh/, /th/) are all bursts of noise that generally last longer than plosives (they have a slower loudness envelope), and they each have their own formant, or filter setting, that changes the character of the noise. Note that /f/ is a fairly even noise sound, while /s/ has more of a sense of some frequencies being louder than others, /sh/ is a more uneven noise sound, and /th/ is a muted noise sound without as much of the upper frequencies.
For the voiced consonants, most of them are essentially the same as the unvoiced ones outlined above, except they also involve the vocal cords, so there is again a pitch of some kind when voiced consonants are spoken. These consonants include /d/ (voiced /t/), /b/ (voiced /p/), /z/ (voiced /s/) and so on. I believe every unvoiced consonant has a voiced version in English (I believe this is also true in Japanese).
There are a few voiced consonants that do not have unvoiced versions and also straddle the line between consonants and vowels. The two closest to being vowels are the /y/ and /w/ sounds. These are basically vowels where the formant or filter is changed while we say them. This is done by changing the positing of the tongue or lips while the vocal cords create a pitch.
Two others, /m/ and /n/, are basically made similar to humming, and the main way we tell the difference is by how the consonant changes to a vowel to determine whether it was an /m/ or /n/. During the transition to vowel, the difference between /m/ and /n/ is similar to the difference between /w/ and /y/.
Finally, /l/ and /r/ are essentially vowels that have rather extreme formants or filters applied do them. They also sound different when they are approached and left (what you might call their formant/filter envelopes).
If you're really paying attention, you've noticed I have not discussed every English phoneme. I have touched on all the musical aspects of phonemes in all languages. Here's more of a breakdown aspects of phonemes:
Those are the primary elements that distinguish different phonemes summarized with musical, rather than phonetic, terms.
For fun, let's break down "hello" and "goodbye" musically, as if we were going to try to make a synth make these sounds:
"Hello"
"Goodbye"
I think you might be best served by linguistics, specifically phonetics.
Pitch is sort of an element, but specific pitch isn't the concern. Instead, some vocal sounds are "voiced" meaning the vocal chords vibrate (producing pitches.) For example, the f in 'fan is not voiced, but when voiced it becomes v like 'van.'
How vowel and consonant sounds are produced is understood in linguistics as a matter of vocal anatomy of the tongue, palette, etc. and described with terms like fricative, labial, etc. There is a complex mapping of the inside of the mouth in linguistics.
You could describe the actions of the voice with acoustics with terms like amplitutde, wave form, etc. But, linguistics actually has a whole branch devoted to the study of vocal sounds.
By the way, in voice training these topics are called diction.
Other answers have pointed out that timbre is the essence of the answer here. However, if what you've noticed is a lack of musical terms to describe timbre, then i think you're right! Music theory doesn't really provide terms to describe timbre in general with the same conciseness that it can describe pitch and rhythm.
Part of the reason for this is that one of the main use cases for music theory concepts over the years has been to provide performance instructions - and with traditional (acoustic) instruments, the most efficient way to do this is simply specify the instrument with which the part is to be played, and add any other performance instructions to the score.
With human voice, of course, the lyrics are also added to the score! So an accurate, if trivial, answer to your question as to why hello sounds different to goodbye could be "the lyrics are different"... though that doesn't seem to be the spirit in which your question is asked.
