High inrush current is handled by a combination of contact pressure and other mechanical design attributes plus metallurgy of the contacts. In the case of relays with a high-inrush version where all other specifications (coil power in particular) are similar, chances are that the only difference is the contact materials.
The main issue with inrush current is that an excessive current can cause the contacts to weld together, so that they do not open when the coil is de-energized, which is very undesirable. Sometimes when this happens you can tap the relay and they will open, only to weld again the next time the load is switched on. Wear is an important secondary concern. The contact materials that resist welding are not typically the best for switching other types of loads such as resistive or inductive.
In the olden days cadmium oxide alloys (AgCdO) were used, but these days less toxic materials such as Silver Tin Oxide (AgSnO2) and Silver Tin Indium (AgSnOInO) and Silver Copper Nickel (AgCuNi) are used. Here is a document from TE which describes a plethora of contact materials.
Contacts that are optimized for high inrush current handling will likely have some other facet of performance or cost that is sub-optimal (such as breaking ability for inductive loads) so there is no one contact alloy that is good for every application. Note also that if you whack the contacts with huge inrush currents, they will get hot from I^2R heating and if you repeat the performance without enough cool-down time, the relay can be damaged.