I don't give a damn for it.
I don't give a dime for it.
Do they have the same connotation?
I'm not a teacher.
I know for sure that you are a very perspicuous person.
Here is a link which gives an answer to your question:
not worth a damn: Information from Answers.com
Here are also a few words concerning the small and great things:
The greatest calf is not the best meat.
Many a little makes a mickle.
Penny and penny laid up will be many.
Grain by grain, the hen fills her belly.
Oaks may fall, when reeds stand the storm.
A small leak will sink a great ship.
A straw may show which way the wind blows.
Great without small makes a bad wall.
Little drops of water, little grains of sand,
Make the mighty ocean and the pleasant land.
I've only heard "I wouldn't give you a dime for it." Meaning that it is worthless to you.
"I don't give a damn" meaning that you don't care about it.
You could say "I don't understand what all the fuss is about the fact that Michael Jackson died. People die all the time, I just don't give a damn about it."
Or, if you see a very expensive painting which does not appeal to you, you could say "I wouldn't give you a dime for that, even though someone paid a million $$$ for it."
I am not a teacher.
I remember reading somewhere that it was originally 'dam', which was an Indian coin of little value.
The other expression about a "dam" is "It's not worth a tinker's dam"
A tinker was a person who used to go around from house to house, and repair cooking pots which were generally aluminum years ago. The little bit of metal that he used to repair the pot was called a "dam".
I remember someone coming around in Germany when I was little. He had a cart that he pulled by hand. He would ring a bell as he went down the street, and he sharpened knives and scissors, repaired pots, and bought and sold rags for cleaning etc.
I am not a teacher.
I was going to say the same thing as Tdol, about the small Indian coin. I believe it was in Language Made Plain by Anthony Burgess, or The Story of Language by Mario Pei.
In the old Amercan Western movies, they often talked about a "plug nickel" as in a fake one. That possibly is also a reference to the small worthless coin.