For bartending, bar tools are the essentials. Without them, you'd struggle to make a proper cocktail. Bar accessories, on the other hand, are supplementary. You can still mix drinks without them, but the experience may feel a little incomplete.
That said, the line between bar tools and accessories has become increasingly blurred, and people often use the terms interchangeably. This is partly because when most people think of a home bar, they immediately think of mixing cocktails — but a home bar isn't just about that. You might simply want to pop open a beer, or drop a few stainless steel ice cubes into your whiskey to chill it down, and that's exactly what accessories are there to handle.