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Can anyone explain what the difference is between accidentals in brackets, and those not?

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  • courtesy accidentals are printed for beginners - as they sometimes don't know the rule that they reign for only one measure or on tied notes they cross the bar ... or just to remember the player ... – Albrecht Hügli Aug 10 at 12:40
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    Worth stressing that the question is talking about round brackets, AKA parentheses.  Square brackets mean something else.  (Editorial annotations, usually; though editorial accidentals can be put above or below the note instead.) – gidds Aug 11 at 10:35
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The accidentals in brackets are called courtesy accidentals. They are used in spots where a player may forget if the note is natural, sharp or flat. They are essential a reminder to the player to avoid common mistakes. For example if a note in one octave has a sharp for the measure that same note IS NOT sharped in any other octave. This is confusing and something players may not see a lot so a courtesy natural sign may be used to shown the player exactly what note to play. They may also appear after a key change or in a measure where a note automatically returns to its value in the key signature after being sharp or flat in the previous measure just as a reminder.

They could be omitted and there should still be enough info to play the piece correctly. However, if you read them you are also guaranteed to play the right notes.

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    The idea that an accidental only applies to the octave where it is written is fairly new. Therefore you better write a natural sign in any other octave or a sharp or a flat depending on what you want, so it is completely clear. – Lars Peter Schultz Aug 10 at 16:37
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    Also very common in choral music where an accidental may have been applied in a different voice. – chrylis -on strike- Aug 11 at 3:10
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We assume that, in the previous bar, C and D were modified by accidentals. (Context suggests they were probably flattened.) A barline cancels out such accidentals, so the naturals on C and D are not strictly required. They are included, in brackets, as 'courtesy accidentals'. The D♭ and E♮ ARE required, as they differ from the key signature and have not already been stated in that bar. So they require no excuse and are presented normally.

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The accidentals in brackets are courtesy accidentals: ignore them and the music won't change.

The accidentals without brackets cannot be ignored--if you do ignore them, you're playing the wrong notes.

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