Before I answer, we should note that Schenker himself was not the best at expressing his own theories. His treatises are at various times rambling, political, polemical, and utterly contradictory. Furthermore, his theories developed throughout his life, so the theories expressed in a 1910 publication are not the same as those in a 1935 publication.
Partly because of this, there are differing traditions within Schenkerian analysis. A Schenkerian analyst who studied from a handful of schools in New York, for instance, will graph a piece very differently than someone who studied at North Texas. (And, needless to say, a Schenker skeptic would graph it more differently still.)
All this to say that there is some level of opinion inherent in this answer, but I'll give what are in my opinion the three biggest strengths and aims of Schenkerian theory.
Showing the Counterpoint
Roman-numeral analysis shows the harmonies created in various vertical columns of a piece of music. But these "salami slices" of chords often miss larger contrapuntal patterns. Is that really a I11 chord at the cadence in a Mozart piano sonata, or is that "I11" really just the byproduct of a delayed resolution of V over a pedal tonic?
Often you'll hear theorists bash a viewpoint as being too "vertical" in approach, and this is what they're referencing. Instead, we should strive to understand the counterpoint—the "horizontal" aspect—in creating the music.
Showing Prolongational Structures
In my opinion Schenker's greatest strength was his claim that a harmony is being prolonged even when it is not literally sounding. For a famous example, consider the progression
I–viio6–I6. Even though nothing in that
viio6 has anything to do with tonic—not a single note is the same!—we understand the
viio6 as a contrapuntal chord that is prolonging the hierarchically more important tonic.
This thought process is closely tied with the rise in interest in Gestalt psychology in turn-of-the-century Austro-Germany. As a simple example, Gestalt psychology allows us to look at a dog and say "look, a dog!" instead of "look, millions of prickly hairs!" The same is true in music; instead of spotting every individual chord, we can spot much larger patterns in the piece. (And in my opinion, these larger patterns lead to a more informed and musical performance. But not everyone agrees with me on that one...)
Showing the Generation of the Work
Lastly, Schenkerian analysis shows how we can understand a tonal composition as having been generated from a background fundamental structure called the Ursatz. This is why most Schenkerian analyses have multiple levels: they have a deep background showing the most basic fundamental structure, a shallow middleground with more detail, then a deeper middleground with even more detail, and then the foreground (also called the musical surface) which shows the entire piece.
As for when to use Schenkerian analysis, it is a theory that explains monotonal compositions. If you're looking at a work within the tonal system that uses one (and only one!) over-arching tonality, Schenkerian analysis is a viable analytic tool for that work.
The theory was not designed to analyze pre-tonal, non-tonal, or polytonal works, but plenty of authors have expanded the Schenkerian system to try and fit it for these repertoires. For a sample of some of this type of work, you may want to consult Felix Salzer's Structural Hearing.