I understand that the present values and duration of liabilities and assets are required to be equal to each other under both cases, and furthermore for Redington immunization the convexity must also be larger for that of the assets.
What is the mathematical requirement for full immunization? I feel as though since Redington immunization has more conditions it should be a "better" strategy but it does not seem like it...
Full Immunization has the same first two conditions, but has a different third condition. The third condition is that each liability cash flow is both preceded and followed by an asset cash flow.
Full Immunization implies Redington Immunization. That is, if you have achieved Full Immunization, then you have achieved Redington Immunization. The converse is not necessarily true.
Each strategy has three conditions, but I would argue that the quantity of conditions is an incomplete measure of which strategy is better. Another consideration would be the assumptions each technique requires. Both strategies assume a flat yield curve and only parallel shifts in yield curve. These are very strong assumptions which are rarely true.
Determining which approach is better depends on what one is attempting to achieve. It may depend on risk tolerance and cost, it may also depend on the frequency or timing of cash flows. Remington Immunization may cost less, but it can only immunize the impact of small changes in rates.
The Exact Matching strategy, which is not quite an immunization strategy, can be very expensive but attempts to dedicate an asset cash flow to each liability cash flow. This may suit the needs of someone who is unsure if they will be able reinvest asset cash flows at the required interest rates.
Different situations may call for different strategies.