For the digital output pins, how is LOW different than no connection?
(Edit for clarification: I've assumed in this answer that you're asking about the difference between connecting the relay to ground/LOW vs. connecting the relay to nothing at all. I've also assumed by the word "trigger" that your relay has some kind of digital actuation built-in, such as a transistor, or that it's simply a Solid State Relay.)

If something is unconnected, it is described as 'floating'. This means there could be random voltage spikes in it caused by nearby electromagnetic fields, making it appear to be switching on/off.

Connecting something to ground or LOW can stop this from happening by drawing away any induced currents.