Tennis jargon: why are some matches "not before" a given time instead of just "after" that time?

To me, in this particular case "not before 8" implies "We'll do our best to make it 8, but circumstances may prevent that from happening", while "after 8" implies "I don't know when it'll be, but definitely sometime after 8".

Minor difference, but at least to me personally, the "not before 8" actually makes more sense here - and sounds better.

Note that the "not before" structure doesn't need to occur only when the event follows another event of a similar type. E.g., if one girl is planning to visit a friend, their conversation might be something like,

Katie: "When should I come by?"
Annie: "Not before 10. I don't wake up early on the weekends."

There's a strong emphasis here on the fact that anything before 10 will be unwelcome, but it doesn't actually say when Katie should come (she might come at 4pm, and that would be completely fine), whereas "after 10" wouldn't stress quite so much the fact that an earlier time would be disagreeable.

In that sense, "not before X" could be said to be somewhat equivalent to "X at the earliest".

"Not before" suggests a deliberate decision to limit the time or start time of some event or occurrence, whereas the connotations of "after" are rather neutral.

So in the case of Tennis, saying "after" may give room for people to hope for perhaps an early start, but saying "not before" seems (to me) to mean, "even if a preceding match is shorter than expected or is cancelled or rescheduled, this match will still as early as 8; however, we reserve the right to start late should the situation require." That's just a lot to read.

There's also a lot of tradition associated with sports and sports terminology. It could simply be that that's how it's done.

Edit:

Warning: Hair Splitting Ahead

From a strictly logical standpoint, "after" technically means "a time greater than another time", whereas "not before" means "a time greater than or equal to another time".

In other words, if we're being stingy, the phrase "after 8" doesn't include 8:00 sharp. You'd have to wait for 8:00:01.