I recently resized a background image for a client from 1500px wide to 2500px wide, and the new, bigger, image ended up being about 10 KB smaller in size. How did I do it, and why is this important?
I used a free program called Squeezer, which reduces the amount of binary data in PNG-format image files without losing any image quality. (That is to say, it performs lossless re-compression.) Even for images as large as this one, it runs really quickly (it takes a few seconds, at the longest; most images are done in under a second, in my experience).
It’s important to keep websites’ file sizes as small as possible. For one thing, it makes websites load faster if there’s less to load. Increasingly, too, users are being forced to pay their ISP per megabyte downloaded; if your visitors have to pay their ISPs more just because you didn’t spend ten seconds re-compressing your background image, you should probably reexamine your online priorities. There’s really no reason not to do it.