Classification is the process of arranging something into specifically designated groups, or classes. An example of this are the "classified advertisements" in the back of a newspaper, in which advertisements are sorted into different classes, including "lonelyhearts ads," help wanted notices or sales of different types of items. The phrase "the classified ads" has become synonymous with these sorts of advertisements in the back of a newspaper or magazine.
To codify something means to arrange it into a specific system or set of codes. The word and its derivatives are usually used in a legal sense to describe laws that have been reduced to a central code and then written down to clarify their meanings. In the U.S., Constitutional law can be described as codified, as it is a set of laws that has been reduced to clearly defined code. In the UK, the legal system is described as noncodified, as it is derived from several sources. However, the "statute law" component of the British legal system can be described as codified.
The word "classified" has other uses, also. It can also be used to describe something which is not deemed suitable for general viewing and is reserved for authorized personnel. In this sense, the word is most often used in a military or political sense to protect certain documents. This usage is derived from certain documents being put into different classes -- for example, a class for general access and classes for limited and restricted access -- but has been shortened to simply "classified." Codified is a more specific word and does not have other usage.
While something can be codified and classified, in both senses of the word, the two words mean different things. For example, a codified political document -- like the Bill of Rights or a police charter -- is designed to enshrine the values of and give transparency to politically authoritative bodies. To classify this information in the restrictive access sense would defeat the purpose of the document. However, if the code detailed in the document is particularly long, it may be classified in the original sense, meaning that it is divided into classes of related items in order to make it easier to navigate the document.