From Latin 'nitidus' (shining, clean), via Anglo-Norman 'nett' — neatness was originally a gleaming cleanliness.
From Anglo-Norman 'nett' meaning 'clean, pure,' from Latin 'nitidus' (shining, brilliant, clean), from 'nitēre' (to shine). Something neat originally shone with cleanliness. Key roots: nitēre (Latin: "to shine").
'Net' (as in net profit) comes from the same root — 'neat' profit was clean, pure profit after deductions. 'Neat' and 'net' were once the same word.