low profile means small height and width; A profile is a side view, often defined as 90 degrees from the front or back view. The antenna that is low profile can be mounted flush to buildings and cars. Low-profile antenna often has high input impedance, for example input impedance of "Spiral" antenna is about 200 Ohms. It also has small gain, "Spiral" antenna has gain of smaller than 3dBi.
One way to realize a low-profile antenna design is to use a high impedance ground plane in place of the conventional metallic ground plane. High impedance surfaces (HISs), which can essentially be considered as artificial magnetic conductors, are popular for their wide-spread applications in reflectarray antennas, low-profile antennas, electromagnetic absorbers and polarizers.
Low profile means which is conformal ,has less height doesnt have irregular protrusion coming out of its design and can be mounted on the plane surface
Importance of antenna height is to avoid effect of ground reflected signal on a radiation pattern. But many cases it is not possible to have ideal height. Hence height of antenna (Mounting) is less. This has an effect on gain. It reduces gain and coverage length or directivity gets reduced. Many a cases counterpoise is used to create 'good ground'
concept is special for planar antennas, it's the height between the radiating part and the ground plan. this height is the thickness of an antenna form a side view . when this height is very small, we talk about low profil eor also slim antennas.
Recently, there is a trend to use AMC (Artificial Magnetic Conductor) to reduce the height of antenna from Ground and reduce the effect of Ground plane on the antenna performance; especially its bandwidth
In terms of physical height, One of my experience is the antenna (It was 12 element array) mounted at the height of about 5 time wave length and same antenna mounted less than 2 times wave length, the gain difference observe was about 3 dB
A microstrip patch antenna or a cavity antenna is low profile, a vertical monopole antenna, such as a monopole over ground (for example a blade antenna on an aircraft), is high profile.