My Translation: I never write long letters anymore.
I believe they both mean the same thing but there's obviously some difference I don't know about.
I think they have the same meaning. The basic difference is that 〜ようになる is commonly used after positive verbs, while 〜くなる is commonly used for negative verbs.
Why? Well, negative verbs are morphologically shaped like adjectives, so they have the shorter 〜くなる form available, and that's what people use 99% of the time. That's not possible with positive verbs, so for those people use 〜ようになる instead—for example, 「見えるようになる」.
To get some numbers to back this up, I searched the Balanced Corpus of Contemporary Written Japanese (BCCWJ) using the freely available 少納言 tool. Here's what I found:
なくなる 6527 results ないようになる 49 results
In other words, the short form 〜なくなる is over a hundred times more common!
In contrast, there are 6513 results for ようになる alone. If you search for this phrase, you'll see that the vast majority of the time, it's used after positive verbs.
(Note: To reduce false positives, I typed