Short Message Service (SMS) & Text Messaging (Texting) are the same thing. It is a means of sending short messages to and from mobile phones. SMS was originally defined as part of the GSM series of standards in 1985 as a means of sending messages of up to 160 characters, to and from GSM mobile handsets. Since then, support for the service has expanded to include alternative mobile standards such as ANSI CDMA networks and AMPS, satellite and landline networks. Most SMS messages are mobile-to-mobile text messages, though the standard supports other types of broadcast messaging as well.
The term SMS is frequently used in a non-technical sense to refer to the text messages themselves, particularly in non-English-speaking European countries where the GSM system is well-established.