After it was announced that DSi owners could buy Wii points to use on downloadable games, many had assumed that it would be possible to exchange them between the two Nintendo consoles, but that won't be happening.
Confirmed by Siliconera, Nintendo will be renaming Wii Points as simply Nintendo Points, and they will be used for both the Wii and the DSi. The screen below shows what the new cards will be like, and Europe's standard 2000 point card should follow suit;

Instead of being able to put some points from one card onto one system, and the rest of the points onto the other system, each card's points will only go to one of the consoles. Many are not surprised at this since Nintendo doesn’t have an overarching account system to control digital distribution.

Confirmed items that you can buy on the DSi with Nintendo points include unique applications, and small games from DSWare, the DS equivalent of Wiiware, with point prices of 200, 500 & 800. Some are speculating that retro handhand titles might show up too...