They both mean that the second half of the phrase is counter to the expectations from the first part of the phrase ("It was cold, but I want outside," e.g.).
"And" is not really that good of a translation at all for a beginner. It's quite misleading (although "and" can sometimes be a good translation, as you'll see below). You should think of them both as meaning "but."
The major difference is that が is weaker. To quote
Japanese: A Comprehensive Grammar,
Quote:
[ga and keredomo] [b]oth express a weak contrast between two clauses, 'but'.
a 恋人に別れはあるけど、友達に別れはありません。
Lovers can split up, but not friends.
* * *
a 忙しかったがいい一年だった。
It was a busy but good year.
b 貧しいけれども村一番の働き者。
She's poor but the hardest worker in the village.
* * *
33.2.1.4 Differences
The contrast expressed by ga can be so weak that S1 ga is more like a preamble to S2, translatable as "and" , or rendered just with a colon.
a. 竹内さんは病院に運ばれたが、頭などを強く打ち、重体 。
Takeuchi-san was taken to a hospital; he's in intensive care, having hit his head severely.
b. 両親は離婚するが、母親は作家と再婚する。
The parents are getting divorced, and the mother is remarrying a writer.
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In case you haven't picked up on this yet, けど、けれども、けども、けれど are all the same thing. Just the full word is けれども and the rest are abbreviations.
Also, what reference book are you using? It's really weird to me that every sample sentence you gave uses です form instead of だ form before が／けれども. It's not wrong, but it sounds like a certain type of person.