Welcome to the world of tournament chess sets. So, the main difference between ivory and natural is that ivory is super white, while natural is sort of a cream color. That varies between light tan and beige or actual cream. The reason most players like natural more is that the pieces tend to blend into the white squares with ivory pieces. With natural, the pieces don't blend into the white squares on the board. On the next point of board material, I personally prefer vinyl boards, as I like to slide my pieces into position a lot of the time. You aren't going to slide pieces with a mousepad board or a silicone board. You will need to lift each piece, and set it down on the intended square, which won't be a problem in a slow, thoughtful game, but may impede your progress when you are in a blitz game, or are running low on your clock. Hopefully that helps.
Ivory vs Natural? And also...
I also prefer vinyl boards and natural pieces. I have some plastic sets with ivory pieces. Under fluorescent lights where I usually play, the ivory pieces take on a pinkish cast.

The mousepad boards do plop flat out onto the table first try. I think that is the case with silicone boards also. With vinyl boards, they have a little memory curl as verylate mentioned. That memory curl is very easy to fix, however, if they are good boards. First, it should be stored with squares side out when rolled up. Then, when opening, I lay it on the table, then roll it up tightly the opposite way, holding that in that position for 30 seconds or so. Then, laying it on the table, mine is always perfectly flat. Of course, I prefer playing on a wooden board as well, but who wants to carry one around at a tournament? The bag even has a slot for the clock. Nice, neat and compact.
It is true about the mousepad and silicone boards not curling at the edges.
I use a silicone board with my blitz/skittles set. The board is sort of sticky in that the pieces will not slide at all on the surface. However when playing the casual stuff we usually pick the pieces up anyway. And not only does it lie flat but it doesn't wrinkle or crease at all.
While I do own a mousepad board, I really don't use it. It does lie flat and not curl, wrinkle or crease. However the material on the top is cloth. So the pieces will not slide on it, even though it is not as sticky as the silicone. And with the top being cloth brings up one of the problems with the board, for me. The cloth top gets dirty and stains easily. I have tried washing one in a washing machine, as suggested by sellers in the past. The board just gets funky after that. The second problem is not a big one, but the board is heavier than the silicone. And since the silicone board won't easily stain and can be wiped off with no problem, I use the silicone where the mousepad board would be used.
When it comes to the question of Ivory or Natural for the plastic pieces, I prefer the Natural. The Ivory pieces are just too white and the black of the weights can often show through to the surface of the piece. I have a friend that I gave a HoS Hastings, Ivory set to 12 years ago. I then gave him the same set in Natural about 2 years ago. The Natural set looks better and has stood up better with the wear and abuse he gives a set. I gave him these sets because the one he was using was really beat up, and he is a friend after all.
Hi, I'm new here and have been a long time chess lover my life but have never really thought about getting a decent set till recently (always just used walmart sets). So I'm interested in getting a relatively cheap tournament set that I can just take anywhere with me and play a quick game.
I'm looking at the House of Staunton for a set and I was thinking about this triple-weighted one (I'll probably get a combo with a bag)
http://www.uscfsales.com/chess-sets/tournament-chess-sets/regulation-tournament-chess-piece-and-chess-board-combo-triple-weighted.html
I think I'll get I'll get it with a burgundy board (although Im having trouble telling which is red and which is burgundy in the pic), but I also saw that If i do a custom build on there I have the option of triple weighted black and ivory or black and natural. What is the difference between white and natural? I don't think natural means cream since thats apparently a different option for the different class pieces. Also whatever the difference is what do most of you all prefer? Like I said i want to get the burgundy set if that makes any influence
Edit - if i can squeeze another question here - mousepad/silicon board worth it? $3 extra for mousepad and $8 for silicon