Second image looks posterized - definitely overdone unless, of course, that's the "look" you are working towards.
Remember, while B&W can be a simple as removing the colour from a colour image, more often then not, they are and require much more. Since B&W photographs do not have the initial attraction or impact of colours to draw viewers into the photo, they require stronger composition, tone, texture and local contrast. "Seeing" in B&W is an art in itself.
Early books on shooting B&W, like Ansel Adams' series or works by Fred Picker or Bruce Barnbaum and looking at the works of people like those mentioned plus John Sexton are a great way to learn the finer points of B&W irrespective of the digital manipulations that are now possible.