Looking at the definitions of again and wieder we're able to figure out which meanings are identical and which are different.
1 one more time; on another occasion
1 drückt eine Wiederholung aus; ein weiteres Mal, wie früher schon einmal; erneut
2 showing that somebody/something is in the same place or state that they were in originally
2 drückt eine Rückkehr in einen früheren Zustand aus; drückt aus, dass etwas rückgängig gemacht wird
3 added to an amount that is already there
4 used to show that a comment or fact is connected with what you have just said
5 (then/there again) used to introduce a fact or an opinion that contrasts with what you have just said
3 gleichzeitig, andererseits [aber auch]
6 used when you ask somebody to tell you something or repeat something that you think they have told you already
6 (noch, doch) drückt in Fragesätzen aus, dass der Sprecher nach etwas Bekanntem fragt, was ihm im Moment nicht einfällt
As you can see there are two meanings of again that aren't represented by wieder. In the following sentence you can't use wieder in German.
The cost is about half as much again as it was two years ago.
And again, we must think of the cost.
Side note: There are also definitions of wieder that aren't covered by again; hence, you can't translate wieder to again in every case.
Last but not least: in some sentences German natives would tend to use a synonym like nochmals where again is used in English.
Could you say it again, please? - Kannst du das bitte nochmals sagen.
The last definition is taken from the entry of doch.