Osmosis is the movement of water through a semi-permeable membrane from a region of high concentration to low concentration (of water). Adding salt, sugar, whatever's in cytoplasm, etc. to the water lowers the concentration of water.1 External skin is keratinized - the cells have a large amount of keratin, a hard-wearing protein that resists water. This greatly reduces the rate which water can enter and leave the cells. Mucous membranes, such as those inside the nose and throat lack this keratinization, and can be damaged by water entering and leaving them.
A hypertonic fluid, for example distilled or fresh water, inside the nose will cause the cells to take in water, and possibly explode. A hypotonic fluid, like sea water, has a lower concentration of water than the nasal cells, and so will cause water to be drawn out, possibly killing them. Damage to the cells of the mucous membranes is what hurts. Try opening your eyes (which are also mucous membranes) under water in the sea, and in a river, lake or swimming pool. In either fresh or salt water, your eyes will hurt, due to damage by osmosis. Drowning in fresh and salt water would probably hurt just as bad, but to drown in a fluid with a massively different concentration of water, such as the super-saturated saltwater of the dead sea would probably hurt a lot more.
An isotonic fluid, such as nasal spray or sterile saline attempts to have the same concentration of water as the cytoplasm of the cells inside the body, so no water moves either way. These substances can come into contact with mucous membranes, such as the eyes and nose without damaging them, and thus can be used to convey drugs (nasal sprays), or wash the membranes (eye washes) without hurting. Isotonic drinks will (in theory) rehydrate the body if it is dehydrated, without depleting essential salts.
1 - This is not looking at things the conventional way round - osmosis is usually considered in terms of the concentration of solute in water. Thinking of the concentration of water in solute, however, makes things simpler when considering a range of substances dissolved in the water at once.