The concepts are different, but the similarity is that anything easy to measure is "hard" and anything difficult to measure is "soft".
Soft sciences measure complex things, like human behavior, that are challenging to define in a rigorous fashion. People are more comfortable with simple things, like atoms and subatomic particles and rigid objects, which behave as a simple unit following simple laws. That makes biology a "softer" science -- not because it's easier, but because it's more difficult.
Hard skills are easy to measure: can you write the code that does the thing, quickly? But soft skills like "can you figure out what needs to be done?" and "can you communicate with clients and coworkers?" and "can you write clear instructions?" are hard to measure.
We have a notion that only things that can be measured are real. But that skips over the fact that some things are important, but difficult or impossible to measure. It's poor reasoning to assume that only things that have already been given rigorous definitions are relevant, but it's popular among people who don't wish to practice soft skills.
It would be great if we could appreciate that science, and software, are human activities done by humans for humans, requiring a mix of skills both easy and hard to measure.