For the first 11 levels, the main thing weaver will want from botanist is lots of Moko Grass for hempen yarn. But BTN has to be level 15 to gather it, and WVR wants it starting at level 1. Most materials are better matched on levels than that, so this is one of the very few things I went ahead and just bought rather than gathered. (Luckily it's only a couple gil each, but once you start needing lots of it, even a tiny price like that can add up enough to be annoying if you're still low-level and poor like I was.) In most cases, if a crafter needs a certain material at level X, then a gatherer or another crafting class can collect or make it by the time that's level X. (On some the providing class may need another level or two, but rarely another 14 like in this case. In fact, I can't think of any other example where they're this far off.)
Through the first 14 levels, Botanist is only able to gather from trees, and through this period, it's mainly just useful to Carpenter. It's after level 15, when the ability to gather from vegetation patches is added, that BTN becomes useful to Weaver as well. (From there on, it's pretty essential to both, but if weaver is your initial interest, then the early levels of BTN might seem frustrating.)
Actually if you start crafting early, it doesn't take long before it's useful, because early in the game what's useful is low-level gear that can be crafted quite easily. (I picked it up as soon as adding extra classes was unlocked at level 10, and it was useful immediately.) On the other hand, if you wait until later in the game when what you want is high-level gear, then yes, it takes a long time to level crafting far enough to do that. And crafting is either a money sink or a time sink depending on whether you gather your own materials or not. (Actually, it's a time sink in either case, but if you gather your own materials, then you can make it free in exchange for being a much greater time sink.)