An under eye serum is just another reference to an eye serum. Some companies like to market their products as an under eye serum as it is more targeted to the exact area of skin that is being treated withe product. While all eye serums are applied under the eyes, some eye serums are used around the orbital bone as well.
In any case, an eye serum that is applied to your orbital bone area will always work well under your eyes. However, on the contrary, an under-eye serum will not always work on the upper eyelid (orbital bone) area with the same effect.
Be sure to know what area you are trying to treat before purchasing any anti aging eye serum.