Well dofollow is just simply a term to state that it is not a nofollow link, since bots automatically follow links even when there's a dofollow term or none. And on nofollow, spider bots just stop going through the link. it's really simple. =)
Dofollow links are counted by Google as backlinks. Nofollow ones aren't. However, they can still drive traffic to your site. Don't underestimate what nofollow links can do.
Follow helps on backlinks to your website. Nofollow make this link like a banner, it's not a backlink, just a reference, where google will not index it like a backlink, propably not show this page when searching your website.
google bots or crawler doesnt index nofollow links. in short nofollow links are useless in google but there are search engines that consider nofollow links.
Only google differs Dofollow and Nofollow
Nofollow links are not considered by google since dofollow links are considered by google and the websites which have good amount of dofollow links may get high rank in google.
In short, do follow means you will get benifit of backlink from google and no follow mean you will not get benifited or your link can not be read by google.
No follow is a Html code attached to hyperlinks to give the bots instructions not to follow to the other page and not include the link in PR calculation for the site..
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nofollow is an HTML attribute value used to instruct some search engines that a hyperlink should not influence the link target's ranking in the search engine's index. It is intended to reduce the effectiveness of certain types of search engine spam, thereby improving the quality of search engine results and preventing spamdexing from occurring. DoFollow = Follow this link.
I myself believe backlinks, whether do-follow or no-follow, are all important. Of course do-follow links are better, though building no-follow links is never a bad thing. As mentioned above, some search engines put more "weight" in the follow tag whereas others do not. I would reveal my research here, but I'd have to kill you... ONLY KIDDING!
The difference between nofollow and dofollow is in no follow the syntax rel="nofollow" added but in dofollow it is not. and search engine give weight age to dofollow.