Balayage VS Foilayage
Another classic example of confusing hair terminology... let's break it down.
Balayage is a technique in which the lightener is hand painted onto the surface of the hair. It is traditionally processed "open air" meaning no isolation of foils or plastic wrap. This technique is meant to mimic what the sun would do to your hair. It is loved for the easy maintenance and no grow out line. As the trend evolved clients wanted more, they wanted blonder and brighter. In comes foilayage. It is a word us hair stylists made up to explain a technique that mimics the natural lived in vibe of balayage but allows the hair to get brighter and blonder. The foils allow a greater lift when the lightener is incapsulated by them because of the heat conduction. The foils can be blended into your natural root, or you may get a "root shadow" painted on that will deepen your base, blurring any lines of the foils even further.
If you aren't sure which is right for you your stylist/colorist will always help you out, some stylists prefer one over the other. Personally I prefer foil work because I feel it gives me more control over the end result.
Leave any questions in the comments below I'm always here to help!