Do they not take advantage of the extra cpu power of the dsi?
There are some pieces of DSi homebrew and exploits to run it. The vast majority of homebrew will not though, and most of what did can be replicated with a DStwo anyway.
To answer the question of the OP though
DS and DS lite are functionality identical save for the screen brightness selection. You have easy wifi, though you are limited to no or WEP based security if you do.
Loads of great native homebrew, ports of old games, and some nice enough emulators up to the 16 bit era (SNES is a bit suspect and you have the usual list of things that will never happen
https://web.archive.org/web/2014112...index.php?title=SNES_games_with_special_chips but megadrive/genesis is more than playable).
http://gbatemp.net/threads/links-to-various-gbatemp-features-over-the-years.352851/ is not a good list of homebrew but has some nice things seen in DS homebrew.
You can enhance DS homebrew beyond baseline flash cart in four ways.
1) RAM expansion pack.
2) iplayer
3) ismart MM
4) DStwo, and not the DStwo+ I guess.
1) is covered in
http://wiki.gbatemp.net/wiki/3_in_1_Expansion_Pack_for_EZ-Flash_V#Third-Party , actually has some nice stuff though nothing most would miss horribly and not everything that was ported to that made it to the others. There are other expansion packs that work (lick's RAM api if you want to go looking for info) and can even play GBA games but the 3 in 1 is known, readily available and generally well liked. It is also the only way to natively run GBA games on the DS (note that the browser RAM pack will not do this), everything else is emulation and without something else from the numbers in this list the emulation of the GBA on the DS is not great, with said numbered things it is hardly flawless but enough to play the bulk of the GBA library.
2) is mainly media player and GBA emulator
3) is mainly media player, gba emulator and a somewhat suspect port of a game focused handheld version of linux called dingux, it has a lot of emulators including a PS1 emulator which you might be able to smack hard enough to get 15 or so fps out of. A flash cart seller purchased the specs to the iplayer and ported the DStwo kernel to it. They go cheap if you can get them and are not a bad way to have an ehanced flash cart, though the GBA emulator is inferior to the DStwo.
4) You can read elsewhere are the wonders of the DStwo. It has a pretty nice GBA emulator, a media player, a far better SNES emulator than the raw flash carts of the DS, a few other nice emulators, a version of DSx86 which does more CPU types and some nice features that many other carts do not have. Other than nailing the battery a bit and costing quite a bit it is the superior DS flash cart, however if you just want DS games then you have many other options and many of those include nice features too. I am not sure if dingux has now been ported to the DStwo, it is certainly not something people really tout as a feature either way.
DSi then. Wifi can be a bit annoying for normal DS homebrew but still possible. No GBA slot means no RAM expansion homebrew. Some find the larger screens of the DSi XL to be slightly pixely but others like it, much like laptop keyboards it really is something you have to go on yourself.
http://gbatemp.net/threads/dsi-downgrades-hacking.393488/ is what I will link, there is more but that covers much of what there is to know. There is not a lot to use the extra cameras or anything so it is mainly a small speed boost for a select for pieces of homebrew.
Nice enough and worth thinking about if you have a DSi but not worth buying a DSi for unless you want the screens.
3ds. Three main flavours in the 3ds, the 2ds (basically a 3ds but without a hinge and no 3d) and the new 3ds/n3ds. The n3ds is a more powerful device, has kind of different exploits to the original 3ds and this is also reflected in the games it can run and some speeds in some games. Technically the DSi also had this with some commercial games but bugger all really made any use of it.
With a DS flash cart that works (the 3ds blocked DS flash carts but various ones bypass it and there is also stuff like
http://gbatemp.net/threads/3ds-flash-cart-unblocker.376719/ ). DS backwards compatibility is fine software wise but some wonder at the screens here.
The 3ds can be hacked relatively easily, and possibly via a software mod in many not terribly uncommon scenarios. I would not call the 3ds scene terrible cohesive, especially compared to things that happened for other consoles in the past (xbox, wii, 360, even the PS3) and present devices (see phones and tablets) but it is getting a bit better and can get things done either way.
Native 3ds homebrew. In my opinion not a patch on DS homebrew and given we saw the rise of IOS (now android is the dominant platform) kill off DS homebrew very quickly I am not expecting anything of great merit from native 3ds homebrew. In my opinion this is also reflected in the 3ds game library where all the devs I liked on the GBA and DS (
http://gbatemp.net/threads/links-to-various-gbatemp-features-over-the-years.352851/ covers the sort of things I like) left the 3ds to go do good stuff on andrios instead. Some seem to be able to enjoy themselves somehow though -- I never cared much for Nintendo first party/captive dev stuff aside from advance wars and that is nowhere to be seen on the 3ds, you may feel differently as first party is kind of moving along even if it is without any stunning things if you have played any of their back catalogue before.
Theoretically the 3ds is probably enough to do the 16 bit era in emulation quite happily, though a lack of interest (you can have a far easier time on android which is where many people have gone) means I do not hold out much hope. Theoretically I guess you might also get some PS1 emulation happening and maybe a sniff at a couple of N64 games but that would require dedicated devs to put a lot of effort in which I am not expecting to happen.
http://wiki.gbatemp.net/wiki/List_of_3DS_homebrew_emulators has more. RetroArch is probably the main focus of emulation on the 3ds for a lot of people, though blargSnes is definitely worth a look. Others have taken to abusing the emulators Nintendo uses for its games to pull things off. Personally I would put them considerably behind any DS homebrew equivalent, and that is also without taking into account any extra features like rewind, savestates, cheats and filters that they provide, but it can be done without a DS or 3ds flash cart so I guess that is why the kids like it.
The unsaid option. DS emulation on the PC is actually not that bad. It is not quite as nice as GBA emulation is/was/will be, and certainly was not at the time, which keeps
on slipping moving forward so that is not a problem. It is however gaining things like
http://gbatemp.net/threads/hi-resolution-ds-emulation.364549/ which do nice things.
So yeah from where I sit the 3ds is just apathy inducing so if you wanted to get a DS or DS lite and a flash cart (anything really, though try to get a half nice one if you can) I would feel comfortable in saying you are not going to be missing out on much of anything, especially if you have some other means to play things that were ported to the 3ds. At the time the GBA and DS represented a superior option for handheld computing but android has just been kicking the 3ds in the balls for years and Nintendo is doing next to nothing to mitigate that. There might be something to it in a couple of years when it is all dead and people are letting modded 3dses go for next to nothing but it is not like the DS where at least a handful and often a dozen or more good games were coming out every month for a couple of years. At the same time I am not sure I can suggest you go all in on android instead -- there is still something to the DS.
/flamebait
