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