From what I can tell using firefox and paint.net, it has a lot to do with how the picture is rendered. Let's take this page for example. If you use chrome, you can right click on a picture and choose inspect element to see what the HTML is for that object. Let's look at some examples:
Super User Logo at top of this screen:
<a href="/">Super User</a>
This code tells us that the Logo itself is not really a logo, but text with a background image that is generated with CSS. When you drag and drop, the photo editing system thinks you are trying to paste in text resulting in failure.
Your user icon on this question:
<img src="http://www.gravatar.com/avatar/91234f2535506a6e532bff242ab2c4ad?s=32&d=identicon&r=PG" alt="">
This now looks like an actual picture with a full HTML address for the photo software to go looking for. Drag and dropping this item works because there is an actual image at that address.
Google Image searchm first result for "Test"
<a id="rg_hpl" href="http://www.google.com/imgres?hl=en&safe=active&gbv=2amp;
On this site, the images are not actually images. The objects are links to other websites. While the image URL is in this object, your photo editor has no idea what to do with it.
While I may be wrong to the actual reason, this is how my understanding of web technology explains this behavior.
I should note that most images are cached locally on your computer when you view them.
Hope this helps