The overture is so telling of the opera's Romantic tendencies, beginning as it does with the depiction of a violent storm. One cannot avoid the impression of a great ship being tossed about in the waves and wind, blowing as it does from virtually "every page of the score" (so said one of the early conductors of the work). This storm is meant to be heard as metaphorical as well as real, however, since turbulent conflicts exist within the souls of at least two of the major characters, the Dutchman and Senta. And given the context of the story, no better metaphor exists than that of the original maelstrom that caused the Dutchman's problems in the first place.
Much is made of Wagner's leitmotif system, a musical device he used in later operas to bring unity to these sprawling works, but there isn't really a leitmotif system at work in Dutchman, as much as some commentators would like us to believe (The German word Leitmotiv, by the way, was not a word coined by Wagner himself but by some later commentators on his musical process). Musicologists of the first half of the twentieth century tried to attach a name to every theme in Wagner's works in order to show what a genius he was in providing a special musical theme or motive for every character, object, emotional or psychological state present in the librettos. However, simply the fact that the composer provided leitmotifs for various elements in his operas is not a sign of Wagner's genius. What he did with those leitmotifs is what made Wagner such an outstanding composer, the way these themes developed along with the psychological development of the characters in these operas. Der Ring des Nibelungen, for example, is fascinating because there is a vast amount of character development across its span of four separate operas. Along with that story development is a musical development of the leitmotifs so that we have, in a kind of musical short hand, a clear idea of what is going on with each character as the story progresses.
Such is not the case with The Flying Dutchman. Wagner's system of related motives unifying the musical drama had not yet coalesced at the time of the writing of this work, therefore any attempt to establish the identity of these themes with any certainty is ultimately doomed to failure. We can, however, say that some of the themes tend to be associated with certain characters or objects in the opera. For instance, the broad horn theme that is stated at the beginning of the overture is more or less associated with the character of the Dutchman, as it appears most often when he is either on stage or when he is referred to in the text. After the initial bluster in the overture subsides, everything dies down, there is a full stop, and a lovely new theme appears in the winds of the orchestra. This theme is associated with Senta because, again, it appears whenever Senta is on stage or referred to in the text. But is the theme actually a 'signature' tune, or is it evocative of her love and devotion for the Dutchman? Or perhaps the theme is broader than either solution and is really expressive of Senta's desire to be the Dutchman's salvation, or the Dutchman's desire to be saved by her. We'll never really know.
What we can know is that unlike the leitmotifs in the Ring, Wagner does not musically develop these thematic ideas. The Dutchman's theme referred to above is exactly the same at the end of the opera as it is at the beginning of the opera. The same can be said for the theme associated with Senta. There is no real musical development along with the dramatic development of the characters in their search for solutions to their problems. This attachment of themes and groups of themes to certain characters, then, can be seen in its infant stage in The Flying Dutchman, a simple precursor to a much more elaborate scheme to be devised for works such as Tristan und Isolde and Die Meistersinger.
Wagner's use of themes in Dutchman is not terribly different from Verdi's use of the maledizione or 'curse' theme in Rigoletto. Verdi simply brings the theme back, either in the vocal part or in the orchestra, whenever Rigoletto refers to the curse of Monterone slung at him by the hapless father at the beginning of the opera. It is a case of what we call 'thematic reminiscence', not a true leitmotif, because like the Dutchman themes, the 'curse' theme never changes. It is the same in Act I as it is in Act IV. It's a clever device, and gives the audience a kind of 'sign-post' to follow through the progression of the work, but it is not a theme with the same dynamic nature as the 'love potion' theme in Tristan or the 'ring' motive in Das Rhinegold.
Wagner does, however, go a bit further in using these attached themes in The Flying Dutchman than Verdi does in Rigoletto, simply by virtue of the fact that there are more of them and they are utilized on a more regular basis and on a more structural level. Indeed one can say that Wagner has devised an entire music drama using only ten or twelve themes that he created to suggest certain characters, characteristics or occurrences in the opera. An entire unified fabric of thematic material tells the entire story in a relatively simple and elegant way. The process may become more complex with later operas, but it is a remarkable achievement even at this early point in the composer's career.