Yes. There is. It is one of the most heavily trafficked site in the Internet. If you have lots of followers, you can easily get traffic to your site.
Sometimes, i do get and sometimes i don't. As far as i know, if there is anything interesting, helpful, useful, entertaining then people will come to our website/page without thinking.
Social Media is one of the best ways to gain traffic, especially Twitter. If you manage good your Twitter account and if you have a good amount of people following you you can get traffic from Twitter.
Twitter can be an amazing source of traffic. You need to have a clear plan of how you're going to attract that traffic and funnel them to your site though. If you just buy a bunch of followers and spam links to your site - you aren't going to get very good results.
There's more people on Twitter that are willing to buy products than there are on Facebook. Facebook users are more interested in socializing and just chatting and don't care about the marketing side of things at all.