I would categorize the ocarina into the following basic styles/categories:
Transverse (Sweet Potato,Zelda Style)
The Transverse Ocarina
The transverse (sweet potato, Zelda) ocarina is the more popular style. If you do a search on the internet on ocarinas you will find many, many more of these types of ocarinas. Why? My guess is because it is easier to make ocarinas in this style, and it is the more traditional style. When played, the hands are held to the side of the head much in the way that the modern orchestral/band flute (Boehm style) is played.
A Zelda Song
There are some parts of the world (Italy, Japan, Latin America, etc.) where the ocarina is a very popular musical instrument, and to the best of my knowledge the transverse ocarina is the more traditional and popular style in those locations.
The Inline Ocarina
The inline ocarina is held straight in front of the mouth, similar to a recorder or tin whistle. In most inline ocarinas, the hands are held side by side (as opposed to one behind the other - like the recorder or whistle).
Ocarina Style Comparisons
Both transverse and inline ocarinas come in variety of shapes, including decorative and artistic shapes like animals, plants, etc. The ocarinas intended to be used as serious musical instruments are usually not the ones in the decorative shapes.
There is some informative insight into various styles and materials at the Master Ocarina Maker site.
Thinking about comfort...
One of the downsides of transverse flutes/ocarinas is that it is difficult to play for extended periods of time.
For transverse flutes, it is not uncommon for dedicated flautists to experience neck pains and problems due to the way that the neck position is held for long periods of time.
It is more difficult to hold the hands to the side of the head than in the front. If you want a quick feel for what I'm talking about, try this quick experiment. First touch both hands to your lips and hold it there for a few moments. Next, hold both hands to one side of your head. The longer you hold each position, the more you'll see the difference between the two ways of holding an instrument.
If you're a Zelda fan...
For video game enthusiasts of the Nintendo game "Zelda, Ocarina of Time", the transverse (sweet-potato) ocarina is the type Link plays in the game.