Difference between DNA and Genome
DNA is any piece of DNA. Genome is a very specific piece (or pieces) of DNA: all of the DNA contained inside one cell, or organelle, or virus particle, from a specific species.

For a species with a single chromosome, like the bacterium, E. coli, then the bacterial genome is equivalent to the cell's chromosome. Same for a virus. For a metazoan, like C. elegans, which has six chromosomes, the nuclear genome is one copy of all the chromosomes and the mitochondrial genome is one copy of the mitochondrial chromosome.

So DNA is a general term, while genome is a specific term that can be used most clearly with a modifier to specify the species you are referring to (e.g., the human genome, or the duckbill platypus genome, etc.).