Here is a collage of two images of two similar looking rain lilies I have. The one on the left - with pointed petals lasts for a day and the one with slightly roundish petals lasts two days in the same conditions. The former is the first to pop up as soon as the rain comes the first time after some months. The second, takes slightly longer. Is there any subtle difference in cultivar or something?

I'm not a rain lily expert, but it might also be different species, besides variation within a species.
Look up Zephyranthus candida and Z. atamasco. They are the most common white ones.

Thanks Leftwood. Checked the Z.atamasco. Mine looks more like the candida variety.