Benefits of a Circular Layout
Circos uses a circular composition to show connections between objects or between positions, which are difficult to visually organize when the underlying layout is linear (or a graph, which can quickly become a hairball). In many cases, a linear layout makes impossible keeping the relationship lines from crossing other structures, deteriorates the effectiveness of the graphic.
In addition to its strength in depicting relationships, the circular form itself has a number of useful properties which are not shared by a rectilinear layout.
The circular form encourages eye movement to proceed along curved lines, rather than in a zig-zag fasion in a square or rectangular figure.
Within the circle, resolution varies linearly and increases with radial position. This makes the center of the circle ideal for compactly displaying summary statistics or indicating points of interest (i.e. low resolution data) which the reader can then follow outward to explore the data in greater detail (i.e. high resolution data).
Also, for a given square area on a page (e.g. square of side `x`), the circular layout can support a larger data domain (i.e. `2 \pi x \approx 6x`) than the square (i.e. `x`). More data can be shown within a given space on a page (or within narrower field of vision).
I've applied circular compositing to represent database structure with Schemaball.