So binaryCIF is newer and is heavily inspired by MMTF.
NGL, the most "loved" JS library (IMO) to show protein fetches proteins as MMTF, while Mol* uses bCIF —Mol* has the official repo of the bCIF format. Alex Rose, developed the former at the RCSB PDB, and the latter in collaboration with the PDBe. He also worked with Antony Bradley, the main author on the MMTF format, who left the RCSB PDB, some 4+ years ago. So there is definitely no competition there.
- The MMTF download of the RCSB works well: https://mmtf.rcsb.org/download.html
- The MMTF format did not catch on —okay, mmCIF is only slowly replacing PDB, which is still the only accepted format for analyses—, hence the need for a bCIF format
- The libraries to do the conversion aren't without glitches
- In the distant future bCIF will be the sole binary format
- mmCIF <--> bCIF conversion is easier
However, I really ought to stress that mmCIF is a better choice to bCIF or mmTF. And even then many tools still hate mmCIFs and use PDB format.
If memory footprint is not really a problem, then stick with mmCIF. If you are running a webserver, you may be sending data as gzipped making this a moot point.