The property that most sharply separates mild steel from alloy steel under high-temperature loading is that alloy steel retains a higher mechanical strength at elevated temperatures than mild steel.

Mild (plain-carbon) steel derives its strength mainly from a ferrite–pearlite microstructure. As temperature rises above roughly 300–400 °C, this structure loses strength quickly: the ferrite softens, cementite lamellae tend to spheroidise (coalesce into rounded particles), and the steel becomes prone to creep — slow, permanent deformation under sustained stress. It therefore cannot hold load reliably when hot.

Alloy steels contain deliberate additions such as chromium, molybdenum, vanadium, nickel and cobalt. Several of these are strong carbide-formers that produce fine, thermally stable carbides which resist coarsening and pin dislocations even at high temperature. The result is retention of strength, hardness and creep resistance — the very reason "hot-strength" and "creep-resistant" alloy steels are used for boiler tubes, turbine parts, and high-temperature pressure vessels.

The remaining statements are false or reversed:

• Claiming mild steel has better wear resistance when hot is wrong — hardness and wear resistance at temperature are exactly what the alloying carbides give alloy steel the edge in.
• Claiming alloy steel is more ductile at high temperature misstates the point; the distinguishing benefit of alloy steel is retained strength, not extra ductility, and higher-strength alloy grades are generally less ductile than soft mild steel.
• Claiming mild steel has enhanced corrosion resistance at high temperature is incorrect — high-temperature oxidation and scaling resistance are conferred by alloying elements (notably chromium), so alloy/stainless grades outperform mild steel here too.

Hence the single most significant differentiator is the superior high-temperature strength of alloy steel.