When faced with generating a visual representation of information, it's useful to ask the following.
Even if the answers to these questions seem obvious to you, they help emphasize the purpose of the figure. Sometimes authors confuse the notion of "showing all data" with "sending a clear message" — it is unlikely that readers actually want to see all your data. It is better to send a clear message and not show the data, then show the data and hope the reader will arrive at the right conclusion.
One of the ways in which Circos helps is that it slows you down. Circos requires that you think about your data and design its layout before you write the configuration file (there is no interactive interface). This initial process of reflection can be both short and extremely productive in helping focusing your message.
Once you have written your configuration file, it is easy to hide elements, adjust scale (either globally or locally — a unique feature), or apply a different format to your data (visibility, opacity, shape, color and even position) based on dynamic rules.
In other words, you can generate a variety of figures without adjusting either the input data or signficant portions of the configuration file. Using dynamic rules, you can draw focus to data positions and/or values — these rules will apply to whatever the input data set is. For example, a single rule can color green all the glyphs in a scatter plot associated with a value of >0.5.
Because the configuration can be composed of multiple files, you can mix-and-match configuration blocks. This is very helpful when your configuration is largely fixed (data domain, position of tracks and ticks) with minor changes (min/max values of axes, zoom levels, etc).
Circos is perfect for data analysis environments in which data is processed in a multi-step pipeline (typically using multiple analysis tools). It can therefore be inserted into the pipeline to produce one (or more figures) automatically.
The circular layout is ideal for showing how different positions within your data domain relate to one another. This relationship can be quantitative (e.g. similarity) or binary (e.g. is/isn't connected). Circos was initially designed to emphasize these kinds of relationships and therefore has many helpful features in fully illustrating these relationships.
By allowing you to adjust the thickness of the ribbons that represent relationships, the progression and orientation of the circular segments, and apply twists to the ribbons to indicate the orientation of the link, you can clearly show a large number of connections between data points (or positions) without exhausting the capacity of the reader.