ya they are the same, but according to Halopedia, the Spartan II's armor is more powerful than the Spartan III's armor.
this is what they wrote
Spartan II's armor : "the MJOLNIR Armor is currently the most advanced piece of tactical military hardware in human hands. Each suit alone costs as much as a small starship. The brainchild of Dr. Catherine E. Halsey, MJOLNIR is Halsey's attempt to create a powered exoskeleton capable of taking full advantage of the physical capabilities of a SPARTAN-II. The battlesuit is constructed in overlapping layers. It is a sealed system, capable of extravehicular activity or operations in toxic atmosphere. It is hardened against EMP and radiation, and has filters that are completely effective at removing toxins and bacteria from local atmosphere. "
Spartan III's armor : "Semi-Powered Infiltration Armor, colloquially called "SPI Armor", was a powered armor system used by the Spartan-IIIs. Though far more advanced and far more powerful than the ODST Body Suits, it was less advanced and less powerful than the MJOLNIR Armor used by the Spartan-IIs. Its main benefit was that it was drastically cheaper to produce than the MJOLNIR armor.
This armor was designed with an emphasis on stealth rather than raw power. It featured photo-reactive panels that mimic the surrounding textures, which was developed by UNSC scientists. It has been described as "part legionnaire mail, part tactical body armor, and part chameleon." It lacks the energy shielding of the Mark V and VI MJOLNIR armor designs, and a single "splash" from an enemy plasma weapon melts the suit's photo-reactive panels, rendering its camouflage capabilities useless. "