According to Collins English Usage,
Badly has another different meaning. If you need or want something badly, you need or want it very much: I am badly in need of advice; I want this job so badly; We badly need the money.
For this meaning of badly, don't use the comparative and superlative forms ‘worse’ and ‘worst’. Instead you use the forms more badly and most badly.
Is this distinction due to its meaning "(very) much", which already uses more/most as its comparative/superlative forms?