The main differences are:
wget's major strong side compared to curl is its ability to download recursively.
wget is command line only. There's no lib or anything, but curl's features are powered by libcurl.
curl supports FTP, FTPS, HTTP, HTTPS, SCP, SFTP, TFTP, TELNET, DICT, LDAP, LDAPS, FILE, POP3, IMAP, SMTP, RTMP and RTSP. wget supports HTTP, HTTPS and FTP.
curl builds and runs on more platforms than wget.
wget is released under a free software copyleft license (the GNU GPL). curl is released under a free software permissive license (a MIT derivate).
curl offers upload and sending capabilities. wget only offers plain HTTP POST support.