Basic Molds have a wooden box to support the liner, with holes in the bottom to help push the liner out. They don't have a lid, so if you gel your soap or want to prevent ash, you'll need to cover it yourself (cling wrap works well).
Premium Molds also have a wooden support box, but add a clear lid that slides into grooves in the mold, creating a seal between the soap and outside air. That seal is the main way ash is prevented, and it lets you watch the gel phase without losing heat. Premium molds also have removable sides for easy liner removal.
Handle Molds come in both basic and premium styles and are built for the easiest possible unmolding. The liner's lip wraps around the wood mold to keep it clean, and handles let you lift the liner straight out. Because of this, the wood box's sides aren't removable (premium style) and there are no bottom holes (basic style), so you simply lift by the handles instead.
Mold Sets bundle everything you need (liner, wood box, and lid for premium) at a better price than buying components separately.
Which to choose? If you don't gel or cover your soap, basic molds are the simplest option. If you gel your soap and want more convenience, premium is ideal. Handle molds offer the same ease of use in either style.