Both 'ordinary' salts and dead sea salts are primarily sodium chloride (kitchen salt - rock salt (funny aside: the bunch of crap translators working for the Dutch branch of Discovery and the likes know nothing about idiom, certainly not science idiom, and apparently have no good dictionaries either (plus looking something up in a dictionary is counterproductive for such translators -- too much time involved, as they are on a per word pay basis), so you see things like 'rotszout' which is a literal translation of rock salt, but has no meaning in Dutch. Even the better paid translators for national TV lack certain idiom -- e.g. they always translate 'statutory rape' in US series with 'verkrachting' (=rape), even though statutory rape in almost all cases is not rape, but consensual sex with a minor, which in the US system has been legally equated with rape), anyhow, as I said, both salts are primarily sodium chloride and will affect lather identically. Salt will kill lather over certain concentrations, as it will influence the solubility of soaps in water, as well as the surface tension effect of anionic surfactants ('soaps').
Dead sea salt contains unusually high concentrations of bromine (sodium bromide), but other than that is fairly normal sea salt. The concentrations of bromine are high enough to warrant extraction of bromine from Dead sea water, but as percentage in the salt, it is negligible.
Hard water contains minerals like calcium and magnesium salts, which are much more deleterious to lather than normal salt, since they actually convert your soap into an insoluble salt, with no residual surfactant activity.
On the other hand, insoluble soap salts make for excellent lubricants. Some high performance lubes are based on lithium soaps.