Well, tonality just means music that is organized around a central tone, generally called the tonic. Most people who use the term tonality also implicitly assume some sort of hierarchy of chordal functions, which are (as the question notes) based on the location within an (asymmetric) scale.
Functional tonality (or functional harmony) is a term that developed out of the theories of Hugo Riemann, who claimed that harmony could be understood by classifying chords into one of three functions: tonic, dominant, and subdominant. All other chords were seen as modifications of these three primary functions.
(Note that the question's idea of seven scale degree functions is a different theory, referred to originally as Stufentheorie or scale-step theory. However, in modern Roman numeral analysis methods Stufentheorie and function theories are sort of blended together.)
Today, music theory often discusses these functions as predominant (with subdominant relegated to one type of predominant) as well as tonic and dominant, but the concept is similar to Riemann's ideas. There are a bunch of general trends in classical style about how various functions are supposed to relate to each other:
- Tonic is often felt as a sort of "goal," frequently used for cadences
- Dominant functions generally progress to tonic functions
- Predominant functions can return to tonic, but their most common function is to move toward dominant in creating a strong progression
- Dominant functions rarely progress to predominant functions (and when they do, it's generally only involving weak dominants)
Now, the problem is that while such constraints tend to describe "classical" style, they don't really describe a lot of modern popular styles very well. From the first blues progressions that saw a V7-IV7 progression, Riemann's concept of "functional" tonality was undermined.
Modern popular styles often have their own logic to progressions (i.e., certain types of progressions are preferred in certain circumstances, but others rarely occur), depending on specific style and genre. These are different from the concepts developed out of Riemann's original idea of functional harmony.
Some people continue to use the term functional tonality to refer to these modern styles with their various harmonic patterns. Some analysts have come up with more schemes, including functions beyond the three I noted above, and they have attempted to describe principles for how these progressions work in various styles. Thus, one might come up with a different type of "functional tonality" to describe other styles. But without specification, the term traditionally refers to its classical roots, with its constraints on harmonic motion.