Like so many other things, crying has the potential to be both beneficial and unhelpful depending on the circumstances.
Crying is a way to express hurt feelings, but by no means the only one. Feelings can also be expressed through words and art, movement or breaking something! There's nothing to say that someone who has a good cry is being more genuinely expressive than someone who writes a poem. I always remember the scene in 'Zorba the Greek' where Zorba danced out his grief.
Compared to other forms of expression, crying is more immediate. But there are, of course, different types of crying. A tear or two may well up in our eye or flow quietly down our cheek. The type I'm thinking of here is a long, deep "good" cry, where we give full vent to our feelings and which is often believed to have a purgative or cathartic effect.
A "good cry" has the advantage of bringing tensions to a climax. It's no longer just an internal struggle. It's become a physical event, and physical events have a beginning and an end. Perhaps one of the reasons we feel better after crying is that it feels as though we've been through something and come out the other side.
On the downside, crying can sometimes reinforce a pain rather than release it. It's possible to remember how something made us cry and by recalling it, we cry again.
We can feel as if we "right back there" with the emotions as strong as they ever were, despite having cried about it before. So as a "cleansing" the effects of crying are sometimes short-lived, or worse, the event can become more strongly associated with crying each time we cry when thinking of it. In these instances, the pain does get better with time. But it's time rather than crying that has moved things on.
The other disadvantage of crying is that people who cry when they don't want to e.g. in a public place, are often left as distressed about the crying as they were about the original tensions.
So overall I would say that crying in a safe environment is a good way to release a built-up pain and tension, but that it may not have deeper, lasting therapeutic benefits.