When I started designing websites in the 1990s, screen size was not an important issue. As long as the site I designed looked OK on a monitor resolution of 640 pixels by 480 pixels I was sorted. Everyone viewed a website on a PC or a Mac and used a monitor whose resolution would not vary greatly from these values. As the early 2000s progressed, resolutions slowly increased and so did the width of the websites I designed. But still, I only ever designed the site for one width and as long as I centred the design, it looked great on a wide range of screen resolutions.
That all changed a few years ago when the iPhone was introduced and more and more people were browsing the web with small screens. Nowadays, there are so many phones, tablets and TV screens being used for Internet surfing that a website has to cope with being displayed on so many different screens and resolutions.
I deal with this by creating a site that can automatically reshape itself according to the screen size. This means that the user always experiences the optimum layout for the device they are using. There is no need for them to use a separate mobile website.