Stygg = Nasty, naughty (bad behaviour). Can also mean ugly in some dialects. Stora stygga vargen = Big bad wolf for example. Dålig = Bad, poor, sick and so on. Examples: Han är en dålig person. = He's a bad person. Jag mår dåligt = I'm feeling sick/under the weather. Det var en dålig film. = It was a bad/crappy movie.

Hi, after some research at SAOL (svenska akademins ordlista), the difference seems to be in that stygg = bad behaviour in general (especially children), a possible synonym is disobedient. Elak (somewhat close to evil) would be then wanting to hurt others. dålig = bad quality of something (taste etc.), being bad at something