Constant means remaining the same but its primary meaning is happening continuously, and it also has a metaphorical meaning of dependable and faithful. So to do something constantly means to do it all the time, as well as remaining constant or the same (and also doing it dependably).
Consistent means done in the same way over a long period of time, including an attribute of fairness and accuracy. It also means being compatible with (as in x was consistent with y). So doing something consistently means doing it in the same way over a long period of time, which does echo the secondary sense of constantly, but constantly also includes a sense of doing it continuously, which consistently doesn’t.
For example, I am constantly taking photos that I put up on social media, every day if not more; I consistently post a books of the year round-up on the first of January every year.