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There are 2 trill notations: 'tr' and 'tr' followed by a wavy line.

What exactly is the difference and when is one used over the other?

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The sign for a trill is 'tr' (in bold italics). The wavy trill line is used to indicate the extent of the trill. The trill line is optional for a single note value, but it's necessary for tied notes.
The wavy line by itself is not enough (sometimes it's misused for other purposes e.g. vibrato).

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The wavy line alone means 'trill', so there's no real need to put 'tr' at all. So, it's one or the other. If there's tr and a long wavy line, that means keep on trilling until the end of that line, where there's often a couple of grace notes leading to the next, un-trilled, note.

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    Sorry to nit-pick, but my perception is that PiedPiper is right; a wavy line with no "tr" is non-standard. But maybe a short one could be confused with the symbol for a turn?
    – Andy Bonner
    Jan 3 at 15:12
  • @AndyBonner - true, I've seen upper mordants which could be confusedly played as trills. And a few trills will end in a turn of some sort anyway.
    – Tim
    Jan 3 at 15:25
  • A wavy line w/o the "trill" sign is most likely to be interpreted as a mordent, and yes occasionally I've seen a "double mordent" in music.
    – Carl Witthoft
    Jan 4 at 18:34

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