I have read multiple articles on the subject of responsive web design vs building a separate mobile site. Like all debates, it depends who you ask but I believe that it comes down to budgets at the end.
The choice is like this, pay for one site with added features to make it responsive as opposed to static (same layout, regardless of size of device) or pay for 2 sites one static and one mobile.
Although it seems like a simple choice, responsive designed layouts can get really tricky and require more planning than static sites of course. The level of extra planning and the level of customization for each device as stated before depends on budget but it also depends on the audience and the type of content. It's easy to get carried away with responsive design when it might not be necessary. The key is balance.
What is responsive web design anyway, you can of course find the formal definition, what is responsive web design, but it comes down to this as far as I am concerned, "adapt a layout to the viewing environment, and probably use flexible images".
Mobile sites on the other hand, deliver content exclusively to the targeted phone device(s). A lot of times this means, condensing the content and optimizing the experience for the mobile device user. Think of a bank mobile site. Try going to www.chase.com for example with a mobile device using the device's web browser. You will find only three options, " Log on, Contact us, and find ATM/Branch.", oh, and a link to their app which is a lot like their mobile site. Chase decided out of all the services and financial products they provide, their mobile users would only care about these 3. I am assuming they didn't just guess, this was calculated based on studies, usability studies and trends.
In conclusion, when you decide whether to choose a responsive web design or building a mobile site in addition to your static site, consider your budget, target audience, and the type of content you are delivering.