Both are sold as unrefined salt, and that's where the resemblance ends. One comes from the Atlantic, gray and moist to the touch. The other comes from a closed lake in the Middle East and is white, dry, and fine-grained. If you have both in front of you and don't know which one to choose, this will clarify it for you.
Where each one comes from
Celtic sea salt is hand-harvested in clay salt pans along the Atlantic. Seawater enters shallow basins, the sun and wind do the work, and the salt worker removes the crystals with a wooden tool. This contact with the clay at the bottom is what gives it its gray tone and the moisture it never quite loses. If you're only interested in this one, we have an entire guide dedicated to Celtic salt.
Dead Sea salt doesn't need salt pans. The Dead Sea is not a sea: it's a closed lake more than 400 meters below sea level, with no outlet to the ocean, and water has been evaporating from it for thousands of years. The salt crystallizes by itself on the shore. It is collected, dried, and ground.
Two different processes, one common point: in neither of them is anything refined. No one bleaches the salt or adds an anti-caking agent.
How to distinguish them at a glance
Gray versus white. Moist versus dry. Irregular grain versus fine and even grain.
What to look for on the label before paying
Three things, and you'll have it in less than a minute.
The ingredient list. If it's more than one line long, something has been added. An unrefined salt just says "salt." Anti-caking agents appear as E-535, E-536, or E-554, and if they're there, they're there.
Who analyzes it. The manufacturer saying their product is pure doesn't mean much. A certified independent laboratory saying so, especially within the European Union, is a different conversation.
The format. 250g bags are very expensive per kilo. If the salt will last you months and doesn't expire, the whole kilo is worth it.
Quick questions
Why is Celtic salt moist?
Because it retains seawater between the crystals. This is normal for hand-harvested salt that is not machine-dried. It has not gone bad.
Do they expire?
Salt does not spoil. The only thing that happens over time is that it absorbs moisture, so store them in a dry place with the container tightly closed.