None, really, at least in meaning. The only difference is that zij is the stressed form (used for emphasis).
If you were doing a listening/dictation exercise, there's a difference in pronunciation:
The 'e' in ze sounds like the e in 'difference' (that sound is called a 'schwa').
The 'ij' sounds a bit like the 'ay' in 'May'.
"Zĳ" is the original form; "ze" is what you can say if the word is not important. Obviously, when the word is important, you can't get away with mumbling "ze", but you have to use the full "zĳ". The audio currently says "Zij is rustig.", without stress, but that's OK: No-one forces you not to say "zĳ". Except that the two forms don't sound the same, so if it's type what you hear, you have to write the right one.
I remember this because of this gif: https://31.media.tumblr.com/771c6cd3adaa06a01fdae292ab7a3510/tumblr_inline_mp0fhkCBaT1qz4rgp.gif. My people skills are rusty