As Mario Party 8 came out in 2007 and Wii Party came out in 2010, it is quite obvious that they listened to feedback and ironed out some of the kinks with Wii Party.
In Wii Party, you play the mini-game *before* you roll the dice, and it matters whether you get 1st, 2nd, 3rd, or 4th. In Mario Party, 1st place gets the coins and everyone else gets nothing and the mini-game in the final round (after everyone has rolled) is pretty much meaningless.
In Wii party, who rolls first is determined each round by the mini-game. In Mario Party, it is determined at the beginning and doesn't change.
In Wii Party, Player 1 hits the A button to cycle through the text when it is the computer player's turn. In Mario Party, the computer takes care of it thereby making it virtually impossible to read the text fast enough to see what is going on.
In Wii Party, they do a great job of putting a colored circle around the A button symbol to denote which player needs to hit the button. It's a nice touch. Mario Party doesn't have that feature.
Wii Party only has one traditional type of game board in "Board Game Island", but they have several other different modes that are much different (though still relying on the same core of mini-games). Mario Party has multiple boards, but that's pretty much it. I like Wii Party's additional game modes, but I do wish Wii Party had more boards to choose from for "Board Game Island" than the one. It doesn't seem like it would be that difficult to add another board layout.
I think the mini-game intro screen which explains the rules and controls of each mini-game is much more intuitive and easier to look at on Wii Party than it is with Mario Party.
Mario Party allows for playing the game in teams. Wii Party has some 2 player co-op modes (like the Friend Compatibility thing), but nothing for the board game style set-ups.