I even make a full game (Pokémon Island) than don't follow this formula, and some players get more than 150 hours of playtime, proving that the formula isn't 100% necessary.
Everyone knows what the formula is. New region with new pokémon and few differences from previous one, Choose 1 starter from a Grass/Fire/Water trio, one or two childhood friends are your rivals (excluding GSC and RSE), challenge 8 Gyms for their badges, defeat the local evil group a team something that everyone use the same costumes wanting to take over/destroy the world using the legendary ancient pokémon (after RSE), beat the Elite Four, and so on. It's been done, and it continues to be done to this day.
Fixed
For me it's VITAL to every game, nintendo or fangame, to make at least, a small change in this formula, like having less/more than 8 GYMs and/or based in other thing than types (something like weather/generation/real animals) something different than E4 plus champion and maybe other quest to do after becoming the Champion.
My favorite games (and the most of pokémon fans) are GSC because of the very good idea of 16 GYMs after beating the E4 and the epic battle versus Red. I also liked the change at the E4 in the Bw. On the other hand, the game that I less like was DPP, because this game plot was almost exactly like RSE, even the final Team battle (and the apocalypse) occurs between the seventh and eighth GYMs.
I really wants that BW2 been Gen VI main games, because playing in a previous region (plus que sequel of BW storyline) had some innovation, but probably Pokémon XY will be another game to follow the formula.
Personally, I wish that the GYM formulas have been restructured after GSC (after the neat use of 16 GYMs). C'mon! There several ways to do a game story when the objective is to be "the best player". Just look some other games like Inazuma Eleven, Medabots, Yu-Gi-Oh! and even other RPGs like Final Fantasy. Imagine if in EVERY Final Fantasy you need to defeat 8 GYM Leaders and in the end need to battle versus an E4?
About the starter: Before the DPP was released, there's a rumor about the trio being Dark/Psychic/Fighting, a good idea to bring innovation. Anyway, the starter is too overrated, its just three pokémon. Some times than I played he official games I just deposit the starter and catch other pokémon XD
The main thing that I liked about the fangames was that you don't have to follow the main games dogmas. You can make a game where the main character is evil, in the space or even making two dimensions! But the people generally prefer to reproduce the main games, including the defects
What formula changes have you noticed in fangames? How much are these changes promoted?
Generally I see about several regions, travel to old places (cof cof GSC cof cof), but in few ones the main goal is to save the world, generally involving a legendary pokémon. These isn't big/original changes, but are changes that make the game more unique. Something I see some interesting ideas like time-travel, universe travel and playing as a old character.
How much are they applauded just for being different (regardless of whether they're actually good ideas)?
The last ones that I cited are very applauded, some are the heart of the games. Unfortunately players generally seek better/custom graphics and online features.
Do you think that these changes are good, bad or just different?
Almost all changes are good because they, at least are just different, that is good =D.
Is there a better alternative to some examples of formula-changing?
Yes, like I said before, using a different number/theme of GYMs/E4 or even using other system like the games that I cited.
Are there some bits of the traditional formula that you'd rather stay the way they are?
The few changes that I don't like are (mechanics examples): The really bad ones, like when you have to pay for get healed in the PokeCenters, stupid evolution methods similar to trade evolution and similar things. The ones than confuses me, like changing the evolution level of some pokémon or the pokémon types (without telling) or the existing types in Type Chart.
Why do you think the traditional formulae are still used nowadays in the official games?
I see two main factors.
Pokémon prints money! If it's not broke,don't fix it! Even if the developers remove several features in the game mechanics and side quests (only keeping the main ones), the series still have high sales. Second factory: The target audience are kids, they generally don't played more than two games, so the lack of innovation aren't too noticed, just like the Pokémon Anime.
Are there features which you've not seen any fangame try to change (and why do you suppose that is)?
Almost all fangames you try to become a regional/worldwire Champion, in a few ones you try to save the world, but, there is more! How about a game that you try to take over/destroy the world? And a game that you try to catch a black-list trainer (mixed with stealth and detective game mechanics)? Or a game with even more focus on catching pokémon? Or even to catch a certain pokémon? Or even you search for the 7 Rayquaza Balls that when you collect all they bring a Rayquaza that fulfill a wish? There several possibilities.
People try the standard with few changes only to fulfill the with of making your own version of pokémon, but this bring games with less innovation and games with less inovation saturates the market. There isn't wrong to make a game to yourself and release it to public, but if you want to have success, and more important (for some people), to your game being still downloaded and played for a better bigger time, your game must be unique rather than a weaker (at least graphics and similar resources) version of official games. This isn't clear in the pokécommunty where almost none projects were completed, but in a more competitive place, this will make the difference (in both ways). More examples: Look at the games that were launched every year (or less time), specially the ones with the game year in the name. How many people will play these games of five years ago or more (like Yu_gi_oh! 2007 or Madden 2008) these days and in the future? And how many will play games like ANY Zelda these days and in the future?
We cannot compete with the official games in technical resources, so let's compesate with good ideas/innovations! This is my view. Several portable games are better than consoles one, specially in the past when these difference are even bigger, the Pokémon RBY is a excelent example.
Do you think there's too much (or not enough, or the right amount of) emphasis placed on being different for the sake of different, and why do you suppose that is?
Being different for the sake of different in Game Development, is a good thing. Differences between the games is that make the fangames library (acervo) more rich ;)
Anyway, like I said before, the pokémon games needs more innovation.
I sense that we have not enough emphasis placed on being different for the sake of different.
In the end, a Pokemon game is a Pokemon game, it has to resemble one. If you want to change it up, you can just create an entirely new game. The same way all Zelda games are essentially the same, all Metroid games are essentially the same and all Super Smash/Mario Kart/etc games are the same. Bread should always remain bread. Changing the general structure, texture, taste of the bread will not make it bread anymore. Same applies to a Pokemon game.
Totally disagree this comparison. I'm not talking about changing to a non-turn based, pokémon capturing humans (lol) or other radical change. Almost all titles cited have more innovation that the pokémon. Every one (except New Super Mario Bros, but for me this series lacks innovation generally have a different number of dungeons/bosses/cups or even themes.
Some games like Zelda 2, Zelda Majora Mark, Metroid 2, Metroid Prime 2 Echoes, Metroid Prime Hunter, Mario Sunshine, Marion Galaxy, Mario Kart Double Dash, among others have its own differences that make these games unique, and I am ignoring graphic factor.
Super Mario World have a different way to separate the worlds
Super Mario 64 has 15 worlds
Super Smash Bros Brawl has a very different story mode.