Which buff is more important early game and why? Which buff is more important late game and why? Does having double ad/double ap make a difference which buff is more important?
First of all, the answer really depends on the champion and the lane you are playing.
Early game & Mid game
During early game, you'll want to get some lane dominance to zone and/or harass your opponent and get more farm and experience. Depending on the lane and champion, you will need different resources.
Top lane : Land of bruisers / melee tanky champions. On this lane, the red buff is a real benefit and makes you a real threat to your opponent. The slow will make ganks easier for your jungle (in the case he is not having a red buff himself) and the damage will give you the upper hand in trades.
Mid lane : AP carries are usually playing in this lane. Most of them make heavy use of mana, which tends to be a limited resource on the early stage of a game. Having a blue buff will give you a huge advantage. The CDR allows you to use your abilities more often to harass your opponent, and the mana regen will negate the low mana pool problems you can have in low levels.
Bot lane : Support and AD carry. Depending on the champions, both buff can give an advantage. The statements for top/mid lane also apply here, but having a 2v2 situation makes the benefit less noticeable than in other lanes.
Jungle : In early game, beside the obvious gain of buffs effects for the jungler allowing for faster cleans and better ganks, stealing a buff from the opposite jungle will put him far behind your jungler, as they provide plenty of experience and gold.
Considering early and mid game, I'll consider that the benefit gained from blue buff is more important that the one from red buff. Having your mid lane winning will help you pressure the other team and get objectives easier (mid/bot turrets and drake).
The red buff still is a significant advantage for a lane, but will usually help more the top lane, which will often have more influence in team fights during mid/late game.
Late game is the time for team fights, and having the team together will lessen the effects of the buff.
Blue buff : Providing CDR & Mana regen. Towards end game, AP carries will usually have some CDR and a large mana pool thanks to items. The blue buff will still benefit them, but not as much as in early game. Note that this is very dependent on the champion you are playing (Ryze will usually not need a blue buff in end game, having sufficient mana & CDR from items, whereas Morgana will use the CDR to be able to cast her binding more often).
Red buff : Slow and damage always stack, and there is no cap to these. This is the reason I feel red buff is more important in late game. Slowing an opponent may allow you to catch him and make a fight a 4v5 or get an easy baron. Having more damage is always welcome.
As for late game, red buff seems to give a more significant advantage. Being able to secure a kill on a bad positionned champion with the help of the slow can force the opponents to turtle or avoid team fights, thus leaving your team free objectives.
Both buffs are pretty important but it depends on the champion you pick.
If you pick a champion that uses mana, I'd go for the blue buff as it regenerates mana and lowers cooldowns, allowing you to go through the jungle quickly without having to go back to base.
(Very good picks for Blue: Skarner, Amumu, Fiddlesticks)
If you pick a champion with energy or otherwise, you could go for either but Red buff in my opinion is mostly used for very early game ganks.
(Good picks for early game ganks with red buff: Lee Sin, Shyvana)
Personally I think blue is slightly better as it allows you to quickly move throught the jungle and get levels. In the end it all comes down to preference though.
It maters what champ you are playing! If you are playing a AP champ like Ziggs blow in more important for magic reg. and cooldowns, but if you are a AD carry like Aish red is more important for the slow and exra damage.
Both are important, but early game, it depends on which Champion. Say you're playing an AD Champion such as Graves. You should get Red earlier. If you're playing Morgana, you should get Blue earlier.
Blue is the most influential buff early game. If you're playing a mana dependent jungle (nautilus, gangplank, fiddlesticks, olaf, udyr, etc.) blue will provide your sustain. Certain champions, if invaded successfully at blue, will be shut down for a few levels.
If the jungle does not need blue (Lee Sin, for example) giving it to mid (assuming they have a mana bar) early makes a world of difference. You're allowing them to harass off the enemy AP mid and slow down their minion farm - which will make a huge difference later.
This is the point where your first real team fight will occur. Assuming the game is fairly even and your ranged AD carry has had decent farm, red is more influential. The early team fights here are always going to be close and the slow from red being pegged by your ranged AD gives you a definite edge over the other team.
Which buff is 'better' highly depends on the situation. Blue is still important so your support can catch up on farm that they let the AD carry have early (get blue -> farm jungle), red is still important for the extra CC and the extra damage from your AD carry.
First of all, what do they do:
Blue Buff: More Mana/Energy Regeneration and Cooldown Reduction
Red Buff: Slows enemy / Deals some Damage over time on hit
Knowing that it depends on your champion. For example if you play a Mana Heavy AP Carry the Blue Buff is the right thing to go. But actually this is mostly situational and depends also on your team and the situation you are in. ( Maybe someone else needs it more, even if you also need it ).
This leads back to a ( in my opinion ) very important part of the game: Always talk with your team! Then you know what the others need and what you can take!