Lime is added to sand & cement mixes for added plasticity in mortars and rendering. It's still used by "old school" tradesmen. Modern additives mostly make this unnecessary, although you'll never convince them ;o)
Lime was used in concrete until burning lime in lime kilns resulted in the production of today's OPC (ordinary portland cement.)
There's no advantage in adding lime to modern concrete mixes. Certainly not in terms of added strength.
There's a lot of confusion over limes.
Hydraulic (or NHL natural hydraulic lime) sets under water. In short, it'll be harder and able to live happily outdoors.
Non-hydraulic needs air to set and harden. Tends to be softer and thus better for interiors.
This is where it gets silly .... Hydrated can be either Hydraulic or Non-hydraulic. Simply put, water is added in the production process to make it more stable and predictable. Modern "Blue Circle" is a hydraulic lime, but it is also hydrated. If you used lime in your projects, this is what you would use.
However .............. after all that...... you really don't need it. No lime will ever be as "white" as "Snowcrete" alone. The type of sand is much more of a colour factor.