I was reading a steel catalog and saw that it categorizes steel sections separately into pipe section, hot-finished circular hollow section, and cold formed circular hollow section. Given the same design strength, for example S355, is there any fundamental differences when designing a section of say, 610mm depth and 10mm thickness, using steel material from these three categorizations?
The geometrical tolerances are usually smaller for cold-formed sections.
The material properties and also be different. For example the final grain structure of the metal will be different, and cold-formed sections usually have higher residual stresses than hot-formed.
Surface finishes like galvanizing can only be done "hot."
This contains quite a lot of relevant material (too much to summarize here): http://www.condesa.com/pdf/en/TUBO_ESTRUCTURAL_CASTV3.pdf
Without more context, one difference between "pipe sections" and "circular hollow sections," could be their intended use. A pipe would typically be used to contain a fluid, and therefore would be have a specification for maximum internal pressure, and tested against leaks. A "hollow section" is just a structural component.
"Pipes" usually have circular sections since that is the most efficient use of material to resist internal pressure, but "hollow sections" can also have other shapes such as elliptical, rectangular, etc.