In the first one, you were supposed to read the letter and talk to John, but you didn't do ANY of both.

The second one is wrong because of time concordance. It could have been:

I didn't read the letter and talked to John.
(= you didn't read the letter but you sure talked to John)

Those sentences are not equivalent. Compare:

I didn't either read the letter or talk to John.
or
I neither read the letter nor talked to John.

Both mean the same as the very first one.

Hope this helps!