In modern usage, colloquia are meetings, most often in an academic setting, in which attendees deliver talks on a topic or set of topics and then answer questions and enter discussion. Symposia are gatherings, not necessarily academic in nature, where a group of speakers discuss a topic, usually before an audience.
Traditionally, a colloquium is a gathering of academics in order to discuss a topic and interact with one another in the spirit of sharing information and perspective with one's peers. Etymologically, colloquium comes from the Latin words for speaking or "to speak" (loqui, -loquium) and the prefix indicating "together."
A symposium, on the other hand, is historically a drinking party in which attendees would revel together and discuss intellectual topics. Symposium comes from the Greek word symposion, a combination of words meaning "to drink" and "together." The most familiar example of a symposium, in Plato's dialogue of the same name, would be the drinking party recalled by Socrates where attendees discussed topics such as love and the consequences of philosophical thinking. As these terms are used in 2014, they are both gatherings of people for the purpose of discussing topics of interest, the salient difference being that a colloquium has a more formal, academic setting while a symposium is informal and social in nature.