This may be a silly question—but, is it illegal for one to enter a Go (or Chess, for that matter) tournament with a notebook in hand?
I can only answer the Go perspective. I hear it is pretty much banned in chess.
Paper notebooks, or preferable kifus, are always allowed. If is not uncommon for tournament directors to supply players with kifus for free or cheap.
I've often seen people use electronic notebooks in Go tournaments to record their matches. However, this is usually accompanied by asking the opponent beforehand for permission (it would simply be rude not to ask). To my knowledge, this is almost always approved, in particular when the display is visible to both players during interaction. Similarly, there is not usually a problem in photographing the board. There have been no real incidents with people cheating yet.
I have never seen it in high level matches, probably simply because anyone beyond low dan level can memorize their matches to a high degree anyway. I usually record my matches after they finished, and possibly take a photo if a strange move happens which is difficult to remember.
In Go, the tournaments with large price money are exclusive to few top players, and Go bots are not able to compete with them, so there is no incentive to use them. This may slowly change in the next several years.
The world chess organisation, FIDE, publishes the Laws of Chess, which are used in tournaments world wide. Only the American USCF uses its own rules, but they'll be very similar on this point.
Recording the moves played is mandatory, except in some situations where there is very little time (blitz games, or when you have only a few minutes until the end of the game). The record is used to check if time controls (like "40 moves in two hours") are made, to help with disputes, and for claims like a draw claim based on threefold repetition of a positions. Tournaments will provide score sheets, or players bring a booklet of score sheets themselves.
Other than that, taking and making use of any notes is illegal:
11.3.a: During play the players are forbidden to use any notes, sources of information or advice, or analyse any game on another chessboard.
You can't take any notes prepared beforehand with you, but you also can't make notes during the game. Beginners are often taught to write down their move before playing it so they can do a last blunder check, but that's an illegal use of notes.
Any electronics and phones are banned from venues (I don't know exactly how far Go programs have come, but chess programs on smartphones play at world class level). However, there is a device called the "MonRoi Personal Chess Manager" which is, as far as I know, allowed by the USCF. But not by FIDE.