The two words mean the same thing. Timid is used more commonly. Timorous seems more old-fashioned, and is strongly associated with the famous poem 'to a mouse' by Robert Burns. It is more likely to be used in literary writing than in everyday speech. Hope this helpps

They are pretty much synonyms, and you can use them in the same situations. OED has "Timorous - lacking in courage or confidence, nervous." "Timid - lacking in courage or confidence." Cambridge Learner's has "Timorous - nervous and without much confidence." "Timid - shy and nervous; without much confidence; easily frightened."