Total Recommendations: 3
These two anime don't really have similar art or styles, however the main concept is pretty similar. They both have a god-solving-problems thing going on; and both gods have problematic pasts. Furthermore they're both assisted by a human assistant, girls with whom they end up developing a relationship which is more overt in Noragami. While Loki Ragnarok focuses on Norse mythology Noragami plays into Shinto folklore, and gods roam around the human world in both of them. So if you're into exploring the relationship between gods, and enjoy a good dash of folklore in your anime, then enjoy!
Makai Ouji and Natsume Yuujinchou are dissimilar in perhaps more ways than they are the same, but there is one striking similarity that I feel makes these two series connected. Differences: The vibes are completely different, as are the protagonists. Makai Ouji has a sort of ridiculous feeling to it, with characters so over the top as to seem unbelievable, as well as a protagonist who is centered around wholly unimportant things (such as his grades). Natsume however has very well rounded characters who in many ways lack the eccentricities of the characters in Makai, but are deep enough to seem real to the point of tangibility. Natsume also has an overarching slice-of-life feeling to it, which differs considerably from the shonen overtones of Makai Ouji. The art and music for both are very different too -- I personally prefer the softer palette and lines of Natsume Yuujincho, with its beautiful watercolor backgrounds, as well as its intriguing and extremely well suited instrumental music; whereas I didn't find Makai Ouji particularly enrapturing musically or visually. The supernatural background for both of these shows is also very different. One deals with very biblical mythologies (If you're familiar with judeo-christian mythology its funny to see the subversions present in Makai Ouji), and the other deals with more traditional Japanese shinto beliefs -- the idea of a 'god' is not present at all. The Similarity: The similarity lies in the main plot elements of both shows. Makai Ouji features a protagonist who is the descendent of Solomon, a great king of Israel who subjugated 72 demons, which has modern day (like, 1800s modern day) repercussions for his descendent/reincarnation William Twining. Because of Solomon's actions, William must deal with the constant incursion of many demons who want his political support, and one demon comes to his aid to protect him from the constant besiegement of these vying supernatural factions. Natsume is similar in that the protagonist, who has the unique ability to see spirits, finds himself the focus of many spirits' negative attention, as he has inherited his grandmother's 'book of friends'. Like Solomon, she subjugated many spirits and bound them to her, and Natsume must decide what to do with this powerful inheritance -- in this he is also helped by a strong spirit who protects him from the more malevolent ayakashi that seek the book of friends. These two anime are definitely more different than they are similar, but despite this they share a strong similarity in plot and the situation that both protagonists find themselves in. Despite this, the protagonists still are completely disparate personalities.
Both have a similar interpretation of the slice-of-life genre: appreciating the beauty in small actions and especially the relationships that develop because of them. Through these small interactions and the main character's gradually developing awareness of the feelings of others, these two anime also both subtly suggest (to me at least) the idea of mono no aware, or the sadness of transient things. The protagonist's character development is also central to both plots; Rekki and Natsume are essentially coming to understand themselves and accept their unique life situations through their experiences with the various characters they meet along the way. They are both very character-driven shows, whose protagonists share a quiet kindness and genuine concern for others. Natsume and Rekki both struggle with their respective 'quirks' (Natsume being able to see ayakashi and Rekki being a Haibane) as these in some ways alienate them from other people and seem to detract them both from a 'normal' human experience. Yet despite (or perhaps because of) these abnormal protagonists and the serene, sonderous lives they lead, both shows ultimately focus on what it means to be human. Also, the setting for both is of beautiful countrysides, rich in color and pastoral accoutrements.