The terms do not have highly fixed definitions, and overlap to a significant degree. However, we can draw a basic distinction based on their technical meaning.

Imperialism pertains to empire. Any instance of one group of people establishing control (directly or indirectly) over another group of people is technically imperialism. Colonialism pertains to colonies. Colonies are settlements established by some subset of a group of people who maintain ties to their "motherland."

They are distinct because one can establish a settlement (colony) in a new territory without imperialism if it is uninhabited territory. Likewise, one can engage in imperialism against others without establishing colonies in their territory. However, the two processes often occur together, and the terms are sometimes used synonymously for this reason. As /u/historianLA notes, the Spanish in the New World engaged both in imperialism (by conquering the natives and ruling over them) and colonialism (by settling Spaniards in the conquered territories, though without attempting to totally replace the natives).

Where the distinction between the two processes may be useful is in thinking about the difference between the establishment of European settler colonies around the world, where the natives were gradually pushed to the margins and the population replaced to a significant degree by settlers (e.g. the USA, Canada, South Africa, Australia, New Zealand - Alfred Crosby's so-called "Neo-Europes"), and places where the Europeans did not attempt to displace the local population with their own settlers, but simply to exercise their rule (e.g. India, Egypt, Indonesia, Nigeria, etc.). The former involved much more "colonialism" while the latter involved much more "imperialism," but all of these examples entailed some elements of both. However, these distinctions are not hard-and-fast, and there's nothing wrong with using the terms interchangeably.