I don't have any knowledge of Sandhi, but I do know the difference between phonetics and phonemes.
If there is a phonetic difference only, then it would mean that the sound is different to the listener, but the difference in sound has no linguistic meaning.
If there is a phonemic difference, then it means that there is a phonetic difference - a difference in sound - but that the difference creates a different linguistic meaning.
To take an often used example, the sounds 'l' and 'r' in English are phonemes - the difference sound produces a difference in meaning. 'leaf' vs. 'reef' for example. However, in some asian languages, these sounds only have a phonetic difference - using 'l' for 'r' doesn't change the meaning, and is perceived as being an acceptable variation of the sound.
Summing up then:
Phonetic difference - difference in sound, but not linguistic meaning.
Phonemic difference - difference in sound and linguistic meaning.
I hope this makes sense.