What is the ¬ key?

¬ is most commonly used to represent negation in symbolic logic; ~ is also often used for that purpose, so it's reasonable that ¬ should appear in place of ~ on a keyboard which uses ¬ at all. I'm not sure why UK keyboards include it, but there's no reason I can see why you shouldn't replace it with ~ if you so please.

[This probably isn't the answer you're looking for, but I'll leave it here in case someone finds this question when looking for meanings of that glyph.]

I have sometimes seen that glyph used to represent a "soft [carriage] return". A "soft return" is a nonprinting character that suggests a line break that is not a paragraph break. See the Soft and hard returns section in Wikipedia's "word wrap" article.

The "Script Editor" app for AppleScript would show that symbol when you'd entered a soft return to break the line to make it more human-readable, but you didn't want to break the line from a syntax point of view (AppleScript was like Python in that whitespace was semantically meaningful).