How to distinguish palmitic acid from lauric acid?
Consider the reaction which happens, what is produced and then what is different about the products? The key has to be in their properties.

The saponification reaction which takes place will produce a salt, in this case it will make sodium palmitate and sodium laurate. These two salts have very different solubilities and this makes a marked difference in the lathering ability. Sodium laurate is higher and thus lathers more readily. 

For more details :

Solubility data

Saponification

Scientific Soapmaking: The Chemistry of the Cold Process
By Kevin M. Dunn