Exome refers to the collection of exons: 5'UTR exons, coding exons, 3'UTR exons in a certain species.
Transcriptome on the other hand is a collection of transcribed mRNAs in a certain cell type or specific tissue. Transcriptome could be comprised of multiple mRNAs transcribed from a single gene locus due to e.g. alternative splicing or alternative promoter usage. Thus, when you download mRNAs databases, e.g. from UCSC Genome Browser, they are usually much larger than the databases containing only the collection of exons.