The words παρακαλέω and παραμυθέομαι are both translated 'to comfort,' but there is a difference between them. The latter word (from παρά and μῦθος, 'a word, speech') in the four places in which it occurs (John 11:19,31; 1 Thessalonians 2:11; 1 Thessalonians 5:14 ) is translated 'comfort' in the A.V., and seems to be expressive of more tenderness than the former.
παρακαλέω (καλέω, 'to call'), which it is difficult to render in any uniform way, is calling upon a person in order to stimulate him to something, it may be to comfort; but it often refers to other things — to exhortation in general, as in Romans 12:8; Titus 2:15; and in some passages may well be translated 'encourage,' as in Hebrews 3:13 , "Encourage one another daily," also in Hebrews 10:25 . See 2 Corinthians 1:3-7 where the word, with the substantive formed from it, occurs several times with a more active force than 'comfort.' In Acts 4:36 the name Barnabas, υἱὸς παρακλήσεως, should probably be 'son of exhortation' rather than 'of consolation.'
An interesting instance of the two words occurring together is found in 1 Thessalonians 2:11 , we are "exhorted (παρακ.) and comforted (παραμ.)"