The form can be altered in both cases to meet the artistic demand. Maybe the arch is square instead of parabolic. Perhaps the demi-arch resembles more of half of a square vs. a cane shape. 

Just like decorating a cake, once the basic form is defined, then comes the icing or, in the case of the balloon artist, the decor style.

Decor styles usually follow two paths; classic decor or organic decor. Again, the formally trained balloon artist will default to envision the 'classic decor' style as the 'organic decor' style is a relatively new trend. The classic and organic styles have many different patterns that may be applied depending on the client's vision, installation environment, and budget. 

Some balloon artists will use a term known as 'faux' organic, giving the illusion of a true organic style but costing less. Some classic decor patterns will mimic an organic balloon decor design too.

The other term that often comes into play is 'garland.' A balloon garland may or may not require a fixed frame. Balloon garlands are also of two types; classic or organic. Sometimes, using the existing architectural elements in the installation environment makes more sense than using a frame with heavy bases, such as with a balloon arch or demi-arch. One benefit is the client may opt to strike or remove the old balloon decor instead of paying the balloon artist to return to the venue and retrieve the framing material. Another benefit is to achieve a backdrop.

Adding balloon decor to your event is a great way to color the space and instantly recall the memory in your photos. Whether you want a classic balloon arch, organic demi-arch, or organic balloon garland, I hope this helps you communicate your vision to your local balloon artist.