This is a fine question, I'd hate to see this unanswered for so long.
Obviously, there's no exact answer to this. And so far, none of the GameFreak officials have decided to bestow their divinity on this question (yet, but let's keep hoping), so we'll settle for the next best thing:
Sprite work for most, if not all the Pokemon related stuff is made by Ken Sugimori. Be it the Pokemon themselves, or the characters, as far as I know, Sugimori is the man behind the game.
Thus, most of the sprites would be drawn, or at least supervised by him. As to why he chose them to be slightly different in various cases is unknown, but it can be reasoned out.
First, let's see what sort of differences occur:
- Change in colour of the costumes:
This one is simple: In the GameBoy and the GameBoy Colour, the colour, ironically was not perfect. Owing to a lack of technological advances that is now enjoyed by the vast majority, the previous generation hand-held consoles were not perfect in colour display. And hence: change in the sprite work.
- Change in design/ skin colour:
Ah, the ever controversial scenario. Racism, skin colour, blah blah. Due to the response in certain parts of the world, Pokemon has to occasionally redesign some of their characters, i.e. make them more/ less darker, or remove certain 'symbols' that might hurt the religious feelings of some people. Nothing very bad, really, but people find a way to get offended, and as a big multi-national corporation, GameFreak/ Nintendo/ the Pokemon Company have to accommodate to the people's demands.
Tricky one here, and also probably your question in general. Why is one standing with his/ her arms open, and the other closed?
One answer that pops to my mind is that the official art/ sprite work by Sugimori lays downs what the game character was designed as. This contains the details of the character, and not the action. This is to portray the feel of the character as opposed to the technicalities of his/ her existence or role. It is to show what exactly developers have in mind.
Now, this is different in the games because, well, a) Characters are more "role-based" in the games. The actions they do are part of the technical stuff. All of this means that they have to coded accordingly. So, just standing there like the official art work may not work in-game where the characters have to move about.
Besides, if most of the actions of the Elite Four Member, for example revolve around giving commands, then during the only time the player sees them, i.e the battle, they can not afford to be portrayed as merely standing.
Thus to accommodate for easier coding and more accurate 'action/ animated' ones at that, it is necessary to sometimes, change the sprite work and the artwork.
Disclaimer: This is purely by conjecture, and the official reasons might differ. Feel free to ask further differences, or point out flaws, and I'll edit them in accordingly!