I am a total ditz when it comes to modding systems. What's that Smash Bros. Brawl + mod?
The Brawl+ Mod uses (or undoes) an issue that Brawl used in order to time the button presses when fighting online. That process is called 'buffering', where as you press a button, the game waits 1/16th of a second (10 frames) before it does it. Melee did not do that since it did not have online, so Melee had more precise control. This is one of the key differences. The other two would be speed (Brawl+ plays a bit faster than sluggish Brawl, but not as fast as Melee appears to) and balancing out the characters. In original Brawl, there is a great deal of unbalance between characters in terms of tourney play, so that some characters will never have a clear shot at a victory due to other top-tier characters overpowering them. Brawl+ attempts to fix it.
Brawl+ is one of a number of projects being done to alter SSBBrawl. The others are "Project M" (attempts to outright clone Melee's gameplay using Brawl), "Balanced Brawl" (attempts to balance out characters), and Brawl- (this project Ubers every character, their motto being 'if everyone is uber, no one is', so everyone has a more offensive moveset).
There are guides to softmodding your Wii, and it's actually quite easy to do so.
With your Wii softmodded, you can do cool things like back up your system to an SD Card, in case something ever goes wrong. If your Wii is ever for some reason malfunctioning, you can buy a new Wii, softmod it, and 'restore' your other Wii's file into it. There are many cool homebrew apps, emulators, and even games for Wii (I have old classics like Quake running on my Wii). Games which normally would not let you copy your save (like Brawl) do not pose a problem as you can use homebrew apps to back up and restore your saves for every game. You can even convert old Gamecube Card saves and save them to your SD Card, or restore them from the SD Card back to the Gamecube Card.
But most people use it to play backups off of a hard-drive. Hell, the app that does that even lets you 'burn' your existing discs TO a hard-drive. I've backed up all of my titles and play them off of the drive, greatly reducing load times and preventing wear & tear of my disc drive (which I had to replace due to overuse; what can I say, I love my Wii).