The GZ file type is primarily associated with 'Gzip'. Ofter found in a double extension with TAR (e.g., .TAR.GZ) where the tarball is the archive which is then compressed using Gzip.
Bzip2 is a free program developed under BSD license that compresses and decompresses files using compression algorithms Burrows-Wheeler and Huffman coding. The compression ratio achieved depends on the contents of the file to be compressed, but usually is far better to compressor based on the algorithm LZ77 / LZ78 (gzip, compress, WinZip, PKZip, ...). In return, bzip2 uses more memory and execution time.