The short answer is that salt doesn't do anything to the asphalt itself. But if and when the sodium ions make their way down to concrete base layer of a roadway our concrete deck of a bridge it can weaken it. If it comes in contact with reinforcement steel (rebar) it will create iron oxide (rust). The iron oxide has a larger volume and can cause the surrounding concert to pop off (spalling). Also, the salt spray can rust parts of your vehicle. Source: I'm a structural engineer.
Asphalt is more prone to expansion and contraction from summer heat and winter snow than concrete. The expansion/contraction cycles make small cracks and pits which make it more porous. Salt turns some of the ice in the winter to water, which seeps into the pores/cracks. When the ground gets below the freezing point of saltwater, the ice expands causing the road to deteriorate.