These two progressions are different, although the second one is derivative of the first. As Ben mentioned, the first dm Bb F C, which is probably best interpreted as vi IV I V, which is a very currently popular progression.
The reason it sounds inspirational is that it encodes a narrative of striving against adversity.
- dm: The initial chord is minor, indicating sadness, trouble, and difficulties. "A" is the melody note.
- Bb:The single change of one note, played out in the melody line ("Bb"), changes this to a major chord --that represents the triumph of will over circumstances. The only thing that has changed here is the attitude of "the protagonist" (the melody line).
- F: There's a brief moment of triumph, represented by the first occurrence of the home chord, but it doesn't sound final, because the melody is still on the third of the chord, not on the root. There's also a nice little subplot here in which the melody note has returned to what it was at the beginning ("A"), but the harmonization has changed from minor to major. The act of will by the protagonist has changed her circumstances.
- C: The chord shifts to the dominant, which both promises an eventual return to the tonic chord, and conveys a sense of resolute preparation. ("G" in the melody.)
Then, by cycling through the progression again, instead of supplying the promised resolution, the song gives an overall message of continuous triumph over unabating adversity --as many times as the world throws something at you, you will respond. The melody line itself couldn't be simpler. A note, a step up, back to the original, and a step down, each note lasting a whole measure. Its brevity, simplicity, extension and rhythmic unity all contribute to its epic feel.
The second progression is the same except for the last chord (A, if we continue to assume the first chord is dm) which is best conceptualized as being the dominant of the vi. So you're replacing the dominant of the root chord with the dominant of the vi. That gives an kind of unsettled feeling In the original progression the last chord is a restorative place to rebuild energy and a promise of eventual resolution. Here, instead, it foreshadows a return of the troubles, giving the overall progression a subtle narrative shift here from "I will always overcome troubles" to "troubles will always return." That's perhaps why this song is called "Restless."
There's a strong argument to be made in this second version for Ben's alternate rendering of i VI III V. The musical theory explanation is as follows: The chord built on the fifth note of a scale is called the "Dominant." In most classical music theory, the dominant is major even when the tonic chord (the chord built on the first note of the scale) is minor. The dominant is considered to strongly lead towards the tonic. The use of a chord that initially seems out of place but that resolves in this manner to another chord is often used to signal a key change. For this, and other, related reasons, the dominant is said to "establish the key". Therefore, even though the rest of the progression is the same, the use of a different chord as the dominant at the end of these two progressions signal that they are actually in two different keys.
