Ask Mr. Smarty Plants

Congratulations on being the recipient of some interesting pass-along bulbs.

Here is a summary of the spiderlily bulb characteristics.

Hymenocallis caroliniana (Carolina spiderlily) 

1.5-2.5 ft tall, fragrant, spider-like blooms up to 7 inches across in clusters up to 6 on a leafless stalk. Blooming March – September. Dormant in the summer.
Leaves pale green, shiny and up to 2 feet long.

Hymenocallis liriosme (spring spiderlily)

1-3 ft tall, fragrant, spider-like blooms up to 7 inches across in clusters of 2-3 blooms on a leafless stalk. Petals form a tube 2-4 inches long. Blooming February to May.
Leaves shiny and up to 30 inches long, 1 inch wide. 

Now the differences: Look at blooming time and the number of flowers in each cluster (on a mature plant) to tell the two species apart.
A posting by J. E. Shields on the Pacificbulbsociety.org forum describes how to tell the two species apart quite well. "Hymenocallis occidentalis (now the accepted name for H. caroliniana) normally blooms in late summer. This may be after the leaves have already died back. It's sister species, Hymenocallis liriosme, blooms in spring  with the first flush of leaves."

The Pacific Bulb Society also have a webpage with many of the Hymenocallis species listed and pictures of the two species. And the Missouri Botanical Garden Plant Finder website has a good description of the two bulbs Hymenocallis lirioseme and H. caroliniana.