"The crowd roared, drowning out the slurping and gnashing noises of the worm, but I didn’t dare a glance over my shoulder. The evernearing stench of it told me enough about how close it was. I didn’t have the breath for a sob of relief as I found a fork in the pathway and veered sharply left."
(Source: A Court of Thorns and Roses by Sarah J. Maas)
Here "sob" is being used like "sigh" - although "sob" is primarily associated with crying, it actually refers to the sharp intake of breath which is characteristic of a type of crying, but can also occur without the crying.
I was too out of breath to sigh with relief.
The speaker appears to be running for his life. If one was terribly frightened, he might be moved to cry at the relief of seeing a way to escape. At the very least, he might let out a single sob. But the speaker is running so fast that he doesn't even have enough breath to sob once.
I don't think there's a specific term for that but you could say "He took a deep breath" and in context it'd be understood as meaning the person took a deep breath while preparing to do something involving significant effort.
It means much the same thing. The only difference, if any, is the length of time a person might devote to expressing relief. A sob of relief, to me, implies a quick, short sigh or gasp of relief, without time for anything more while "tears of relief" suggests the person was more relaxed and could sit down and have a good cry.
Just to add - the passage you quoted referring to "tears of relief" suggests the person was out of immediate danger, and would normally have been able to relax for some time (and cry tears of relief) but decided he couldn't afford to, as he found himself in a passageway that was unfamiliar and that was taking him farther into the labyrinth.