The Full Spectrum conversion (clear glass) makes the camera sensitive to UV, visible, and IR light, allowing switching between various external filters to achieve the desired effect, this allows going between infrared and regular shooting with one camera, however on DSLRs opaque infrared filters will block composition.
The Two Spectrum conversion (IR + Visible) makes the camera sensitive to visible and IR light. It works the same way as the full spectrum, while filtering out UV for better color rendition in visible light shooting.
When to get these conversions: Cameras with these infrared conversions offer the most flexibility to narrow down the specific spectrum at the lens level, allowing you to get the most out of one camera body. The only drawbacks are the requirement for addition filters, and a blocked viewfinder on DSLRs. These IR conversions require the use of live view and are perfect for mirrorless cameras.