As a web developer, I have to cut a layout similar to this (example website by Ruben Bristian):

Should I bother with cutting multiple small images like a logo:

a label:

and so on? Or should I just make one big background image with all elements like this:

and make a positioned <a href> with display: block; for a linked logo?
A single image has smaller size than multiple elements altogether. What are the other pros and cons?
Use separate images.
Here are a few reasons why:
Maintenance:
It’s going to be much easier to maintain in the future, if/when there comes a point when you want to build on what you already have. Furthermore (and subjectively), the background image is not critical to the design. It wouldn’t look broken if parts of the background were clipped. It would look broken however, if the logo were distorted.
Bear in mind also that newer, sharper displays are being developed. It’s much easier to display the standard resolution background (it’s already blurry, so clarity is not essential), and maintain two versions of the logo. One for standard displays, one for HD.
Semantics: What if the user has images disabled? Sure, it’s unlikely, but what about Google? You should have some proper markup with real content. Your site needs real textual content in order for Google’s crawlers to gather information about it. Use CSS image-replacement techniques to build the interface.
Another note on HD displays:
It’s convention to serve larger images to HD (retina) displays, and use CSS to downsize them, effectively increasing their dots-per-inch. If you use just one image, the user will have to download a considerably large image. More bandwidth used by you, and slower experience for your users.
Furthermore, the text will look horrible on HD displays. It makes much more sense to allow the browser to render razor-sharp text to the user.
Accessibility: For a start, screen readers won’t have a clue what your site is about. That might not be so relevant in this case, but it’s best practice to build and accessible website. If you want to include some smaller text on the site, some users may be unable to read it. Normally they would increase the font-size, but if you use images, they’re powerless.
I may have over-dramatised this answer, but the advice is well-intentioned.