The current rating is down to the power dissipation which is dependant on the amount of heat the component can get rid of before it gets too hot.
A low current rectifier diode has thick leads which act like a heatsink and dump heat into the board. A larger rectifier diode will require a real heatsink.
A Schottky diode will generally be able to handle more current than a silicon diode in the same package style because it has a lower voltage drop for a given current.