Yes, a SATA II SSD is faster than a SATA III HDD.
HDDs can be SATA III "compatible", not SATA III "capable". No HDD can spin fast enough to achieve SATA III speeds.
HDDs labeled as SATA III have a small cache buffer (16MB or 32MB or 64MB) in them that can transfer its contents at SATA III speeds, but that's it.
HDD manufacturers label their drives as SATA III mostly for marketing purposes.