What the OP is observing when cutting stainless steel is a phenomenon known as "work hardening". In general, the 300-series (austenetic) stainless steels are more difficult to machine due to the higher chromium and nickel content and the formation of the austenetic structure at solidification. The iron-nickel-chromium alloy forms a structire wherein the iron and nickel have alloyed, and the chromium tends to be depleted in the inner region of the grain, while of much higher concentration at and along the grain boundary, where it forms a complex with the carbon in the steel. The lack of alloying with the carbon, and its preferential deposition along the grain boundaries is why the 300 series stainlesses do not respond to hardening by heat treating. They remain ductle, and high in fracture toughness, which is why they are more difficult to machine than ordinary carbon steels, which have a lower fracture toughness.
Are stainless steels "Stronger"? In general, the answer is no, because the tensile strength of stainless is generally lower than that of heat treated carbon and alloy steels. Since carbon and low alloy steels are normally heat treated to enhance their characteristic physical properties (hardness, tensile strength) and most common stainless steels are non-responsive to heat tretment, it is safe to say, as a general rule, carbon steel is "stronger" than stainless.
There are, of course exceptions. The 400-series stainless steels (ferritic) can be heat treated to alter their native properties, and enhance desired properties. The precipitation hardening stainlesses (17-4PH as an example) can be hardened by heat treating, and then age-hardened further.
The complexity in what properties a steel has, how they may be altered by variou heat treatments, what effect the atmosphere while being heat treated has on the properties, and what post-heat treatment regimen the material is subjected to means that a metallurgical engineer has good job security, and the choice of the proper material is vital to the ultimate function of the item manufactured.