To piggyback on what hadley said a bit, I find thinking of a data frame as a "collection of records", and a matrix as "two dimensional data" to be a bit better.
One useful heuristic worth asking is "Does it make sense to sort this data by something". In that case, you have a data frame. Whereas if you want to perform matrix math on something (inverting it, multiplying it by another matrix, reducing it, etc.), you have a matrix. Things that I use a matrix for can generally also be expressed as a data frame with columns rowId, colId, and value. If it doesn't make sense in that format, a matrix is generally not the appropriate structure.
One useful heuristic worth asking is "Does it make sense to sort this data by something". In that case, you have a data frame. Whereas if you want to perform matrix math on something (inverting it, multiplying it by another matrix, reducing it, etc.), you have a matrix. Things that I use a matrix for can generally also be expressed as a data frame with columns rowId, colId, and value. If it doesn't make sense in that format, a matrix is generally not the appropriate structure.