Spindle speed is the speed at which the drive's memory disc (or platter) spins. A higher spin rate improves both positioning and transfer performance for sequential operations. This means that the faster it spins, the faster data can be read off of the disk for single operations.
The interface is the type of connection (such as FireWire or USB) used by your hard drive. Home systems will mostly use an IDE interface. Professional systems can use higher quality interface, as SCSI.