In the bad example, Jean is too vague, and has a few red flags, like suggesting that her last job didn't end well, and making an inappropriate joke about Scorpios to gloss it over. You can tell people you're great, but it's better to prove it by offering specifics, like she does in the better example. In the second example, Jean's letter body is more clearly organized, and makes sure to give the most relevant details about her career so far.
Also, her closing call to action sounds more like a threat. You want to leave the reader with an avenue for follow-up, but don't intimidate them into responding. After all, they're not obligated to respond. You want to make them want to reach out to you for more information or next steps.