The first Pentium III (Katmai) is basically nothing more than a second generation Pentium II (Deschutes) with added streaming extensions (ISSE):

That's the first Pentium III with Katmai core and 512 kB external L2 cache

It was soon followed by the second generation Pentium III, with the well-known 'Coppermine'-core. Coppermine was first introduced for Slot-1 and was also the first Intel processor to reach 1 GHz, about 5 months after its release.

This one is the Slot-1 version of the current Pentium III with Coppermine core and 256 kB on-die cache, running at full core clock speed.

Only a few months after the introduction of the Slot1-Coppermine, Intel introduced the Pentium III for Socket 370 FC-PGA (Flip Chip Pin Grid Array), which is identical to the Slot-1 version in terms of features and the die. The move back from a slot-processor to a socket-processor was only logical, since the integrated L2-cache of Coppermine made the Slot1 CPU-PCBs obsolete. Intel used the slightly modifiied Socket370, which had already been known from the second generation of Celeron processors.

Today, the Pentium III has returned to the socket due to keep the costs down.

The Pentium III for FCPGA is the only version still available. The older packages have pretty much been phased out. It is available in various versions between 600 MHz and 1 GHz with either 100 or 133 MHz processor bus clock. The fastest type with 100 MHz FSB runs at 850 MHz core clock, while the fastest version with 133 MHz FSB runs at 1 GHz today.