Both SATA-II connectors are backwards compatible with SATA-I, so it's OK to use a SATA-I drive on a SATA-II board and vice versa. The primary difference is the speed of the connection - SATA-II operates at 3.0 Gbps, whereas SATA-I operates at half that, or 1.5 Gbps.
In most cases, you won't gain any speed improvements using SATA-II over SATA-I, unless you're using very fast server-grade disks or SSDs in your system. Ordinary desktop hard drives cannot saturate the 1.5Gbps connection.
Technically, SATA-II is a misnomer for Revision 2.0 of the SATA specification, just as SATA-I is a misnomer for Revision 1.0 of the spec. Revision 3.0 of the spec adds 6 Gbps speeds. All of these are properly referred to as just SATA.