This is a question I have sometimes been confused by, but having now studied both I thought I’d attempt an explanation of the differences, for myself as much as for you!
(Warning: the following post may contain geekiness)
Phonetics – this is the study of speech sounds.
During the first part of my course we had two hours a day of Phonetics. It involves making a lot of strange sounds, ranging from “uoaei” and “daa ada taa ata”, to practising gargling (a uvular trill [ʀ], which I’m rather proud of) and sticking your fingers down your throat to stimulate a constriction of the pharynx (yeah, thanks for that one!).
We learnt (or tried to learn…!) to recognise and produce and transcribe the most common sounds in the world’s languages (and some of the not so common) as laid out in the International Phonetic Alphabet.
I found it all very exciting (yes, I know … geek); it’s simply incredible the range of sounds that can be made with the human vocal apparatus!
What I find really interesting though are the tiny adjustments that can be made, which are recognised as distinct sounds in some languages (but heard as the same sound in other languages)…
Say the English words “pin” and “spin” to yourself. Does the “p” sound the same to you?
I’m guessing you’ll say “Yes”? Well it’s not!
In “pin” the “p” has an extra puff of air after it, known as ‘aspiration’, so this sound is written phonetically as [pʰ].
In “spin” the “p” is not aspirated, so it is written simply [p].
Aspirated and unaspirated sounds are heard by most English speakers as the same sound, whereas in many languages (such as Hindi) they are recognised as separate, distinct sounds.
… which leads us onto …
Phonology – this is the study of the relationships between sounds within a language, or between different languages.
Once you have studied a language phonetically, and noted down all the tiny details of the sounds and the differences between them, you then move to phonology and work out how the sounds fit together in the language.
So, for example in English, although a phonetician would notice that English contains both the aspirated [pʰ] and the unaspirated [p], with phonology you’d see that actually [pʰ] occurs only at the beginning of words, and [p] does not occur at the beginning. Therefore (taking into account other things as well), they are just variations of one “phoneme” /p/ (a group of similar sounds) and they are recognised as one sound by native speakers.