Asphalt deteriorates over time and will require expensive maintenance to keep cracks and weeds and tree roots from making it unusable. Concrete, though maybe more expensive initially, is cheaper over its long lifespan especially if there is any possibility of seasonal flooding. To build an asphalt trail right you have to excavate and put in several inches of gravel or road base, and then 3 or 4 inches of asphalt. This article provides a good description of the process: Porous asphalt shows advantages for trail surfacing.
Unless you're building a bikeway as part of a larger community trail network, most people would use crusher fines. The exception is if you have flowing water along or across it periodically. If properly installed with appropriate crushed rock (not "gravel") it looks natural, compacts well, dries out after rain or snow, is "firm and stable" according to accessibility guidelines, minimizes tripping hazards, and doesn't feel hard underfoot. This has all the facts for design: The art of building crushed stone trails.
Another article giving the pros and cons of different surfaces: Rio Grande Trail Corridor Study: Trail Surfacing Report
And the attached presentation is very informative but geared toward a community trail system and especially greenways along stream corridors.
These and more resources are available in the Building and Maintenance portion of our Resource Library.