I already owned a "normal" Nintendo when I decided I wanted to buy a Nintendo DS Lite as well. Why? Because first of all a lot of things have improved and have become much better compaerd to the original Nintendo DS.
First of all the LCD screen has become brighter and you can change its brightness. Even with the light turned "off" you are still able to see your game, which is a major improvement from the original DS handheld console.
Second of all you only have to recharge your NDS after 19 hours instead of the 10 of the original one (if you set the LCD screen's brightness to zero). This is very nice to know, especially when you're in a car for quite a while and you want something to entertain yourself.
A lot of the additional accessoiries sold are for the Nintendo DS Lite and you will probably not be able to use those for your normal NDS. There you go: another reason to buy the Nintendo DS Lite.
The downside of the Nintendo DS Lite is that it doesn't come with a wrist strap, while the original NDS did. You will have to buy it separately from the Nintendo website. As far as I could tell their online shop is only available to people from the US and Canada, so people living in, for example, Europe, won't be able to order it.
The buttons on the NDS Lite have improved, except they have moved so if you're used to the original NDS, you really need to get used to the new positions of the buttons. The start button and the select button have become really small and almost impossible to quickly press while playing a game, which is annoying. The power button has moved to the side and has improved, so you probably won't accidentally turn off your NDS Lite.
The stylus has also moved to the side and has got a better grip to it. It's thicker and I enjoy using this one better than the thinner stylus of the original NDS. It also fits the colour of your NDS.
Another downside is that you cannot use the adapter for the original DS because it simply won't fit. The handheld console obviously does come with an adapter but it's still annoying. I don't really understand why they changed that.
+ Brighter screen, you're able to change the brightness.
+ Better stylus.
+ You can play games for the NDS and the GBA on it.
+ Sound has improved.
+ You can close the GBA slot (slot 2) when you're not using, so there won't be much dust in it when you use it again.
- Different adapter than the original DS' adapter.
- The original DS is heavier, thus it's nicer to hold than the DS Lite.
- If you already own the original DS, the price of the DS Lite might turn you off.
- Doesn't come with a wrist strap.