• Reply and answer are both synonymous though both are used in different contexts
• An answer is a reaction to a question in writing or speech that proposes a solution
• A reply can be in response to a question or it can merely be a response to a statement.
The thesaurus says this:
1. an answer is something said, written, or done in return; a reply is an answer that satisfies a detail in a question
2. reply is generally used when the answer is understood to be verbal - either written or oral; response, by contrast, has a more general application and may be used for a variety of reactions
3. respond suggests a quick reaction while reply implies a return commensurate with the original demand or question
Here are the definitions, from the American Heritage Dictionary:
answer A spoken or written reply.
reply An answer in speech or writing.