Specifications are actually requirements, but not the other way around. I will explain.
Requirements describe what is wanted from the product, and what is not wanted. Requirements can either be implementable or non-implementable. For example, we want the new website to be developed in PHP (implementable), with an attractive, colorful design (implementable), appealing to the youth (non-implementable). The website should not be boring (non-implementable).
Specifications are implementable requirements that are sufficient to build the product. An example of a specification (in addition to the above implementable requirements) would be "The website has to be hosted on high speed servers in the US", another example of a specification "The code should be documented"