If you are a blogger with couple of dozens visitors a day who purchased a WordPress theme or even uses a free one, then you should not care about it at all. Somebody, hopefully, had already done a required minimum for you to make your blog look okay on major devices.
A responsive web design is a must for the website with lots of groups of users who normally perform different tasks, depending on the group or device it is accessed from. Intranet might be a really cool example, since it often requires intelligent content prioritization to improve user experience.
Basically, UX is the first benefit, especially there really tones of tasks and set of information available. By improving user experience you can either improve conversions or improve task performing speed (e.g. by your employees) or reach any other goal that might be critical to your organization.
Before any design change you should do as deep analysis as possible, because there is a risk of making things worth. Learn how the website’s usage is different depending on the device or visitor groups and what the major tasks to perform are. The next step is prioritizing – you make all the important features easily accessible and usable.
Uniformity between devices is not necessary, don’t forget about it. It is really okay to have different user experience on different devices. Simply put main navigation and elements to the prominent place, then remove all that is really might not be needed at all.