Truly responsive websites don’t just scale for screen size, they rearrange themselves for best effect, highlight what's important and remove what isn't important
Columns shouldn’t just move below the rest of the content on a smaller screen where important content will be missed. Content must be prioritised, with important content being highlighted and less important content moving down the page or being removed
Diagrams can be difficult to read when reduced in size; it's far more effective to present a custom version for smaller screen sizes
Traditional drop down menus are ineffective on small touch screens. Menus must rearrange themselves for the small screen and for touch.
It can be tough to tap the link you want when they're jammed together on the screen. Content, and particularly links, need have the correct spacing applied to work well on small touch screens
More and more people are doing their research using a mobile or tablet device. The traditional website doesn’t work well on these devices, as most are just scaled to the point of being unreadable with links and menus unusable due to hover effects which don’t work, and links which are too close together for the correct link to be clicked reliably.
On top of this Google is now prioritising mobile friendly websites for searches on mobile devices.
A next generation website is truly responsive; not only does it resize itself for the viewing device but it actually changes the way it works so it can be used effectively on small touch screens.