What is the difference between comparative linguistics and contrastive linguistics?

Comparative linguistics is part of historical linguistics and refers to the process of establishing family relationships and reconstructing proto-languages (= ancestral languages). For example, if we compare English, Dutch, and German, we can find a number of cognates, that is, words that are similar in phonetic form and in meaning such as English 'book' Dutch 'boek' and German 'Buch.' If we compare the equivalent words in French 'livre' Spanish 'libro' and Italian 'libro,' it's clear that English, Dutch, and German are similar to each other and that French, Spanish, and Italian are similar to each other. This suggests that English, Dutch and German belong to a language family distinct from that which French, Spanish, and Italian belong to. The former is called proto-Germanic; the latter is known to have been Latin.

Contrastive linguistics is part of applied linguistics and seeks to establish the similarities and differences between a language learner's first language and the target language (= the one being learned) in order to attempt to predict where learners will have difficult and make mistakes. For example, languages like Japanese, Korean, and Mandarin do not have articles ('the' and 'a'). We can predict, then, that speakers of these languages will have difficulty learning articles when they learn English. Such is the case. Note, however, that contrastive linguistics is not as straightforward in its ability to predict mistakes as the example I've given. It is most powerful when predicting difficulties in pronunciation, but many types of grammar errors in second language learning occur in areas that contrastive linguistics cannot explain. For example, Spanish speakers have personal endings on verbs (e.g. duerm-o 'I sleep' and duerm-a 'she sleeps'), but they frequently omit the -s in English present tense forms like 'eats' or 'sleeps.' Contrastive analysis would not predict this problem since the -s in English parallels Spanish inflection.