An important distinction can be made between the context of situation and register of a text. They often seem to be conflated but Halliday does distinguish between the two.
Here is a headline from the satirical magazine The Onion:
Coarse Sponge Excited To Join The Smith Family Dishwashing Team
The humour here works only because we can make a distinction between the context of situation and the language that typically accompanies it – by a mixing of the register of a company announcement (excited to join, the ~ team) with an unexpected field choice (sponge).
Context of situation, as the name suggests, is “the immediate environment in which a text is actually functioning ” (H & H, p.46), or the social and physical environment where the action takes place in terms of its field, tenor and mode. Importantly, this may or may not involve language. In fact, there are certain contexts in which the use of language is actively discouraged or even proscribed, such as Berstein’s (1971) concept of ‘resticted codes’ or the cultural value of silence in Japan (King, 2011). If you’ve ever been on a visit to the Sistine Chapel in Rome, with the attendants continually calling for silence, you’ll see one good example.
Register, on the other hand, is entirely semantic and is the “configuration of meanings that are typically associated with a particular situational configuration [and] include(s) the expressions, the lexico-grammatical and phonological features, that typically accompany or REALISE these meanings” (H & H, p.38-39). Register is a probabilistic tendency for certain items from the semantic and lexico-grammatical (and phonological/graphical) systems of ideational, interpersonal and textual meanings to co-occur in certain contexts. It is the semantic interface between the external context and the internal language of a particular text, and may range from restricted registers (the language of aviation, for example) to relatively open ones (casual conversation) although no registers are completely open as even casual conversation exists within certain boundaries and conventions.