Antennas and Ground Planes
There are many, many different designs for antennas, and some designs are quite unusual.  Antennas commonly use a ground plane, but this is not a strict requirement. A loop antenna and a dipole are two examples that don't require a ground plane.

The basic requirements for an antenna are: 


a good match to the circuit driving it (and almost always resonant
at the operating frequency), so that the most power possible can be
put into the antenna, and
having current flowing along its length, so that the resulting
fields radiate that energy into space.  (Receiving antennas are just
this process in reverse).


Item (2) explains why you can't just stick a small tank circuit on a board and expect it to radiate efficiently.  

Item (1) generally comes under the topic of "tuning", where you bring the antenna into resonance or wherever it was designed to be tuned.  A dipole antenna is effectively a resonant length of wire broken in the middle to allow the feedpoint to be inserted.  A "ground plane" antenna removes half the dipole and substitutes the ground plane for that.  The inductance of the radiating element works with the capacitance between it and the ground plane to form the resonant circuit that gives the antenna proper tuning.  When  used this way, the ground plane may be called a "counterpoise". 

A helical antenna coils up the radiator somewhat, to increase the inductance and shorten the length.  Shortening the antenna affects its performance, as mentioned earlier.

So far, we've got a coiled radiator sticking up above a ground plane.  But they've got a surface-mount version that lies parallel to the board.  I can't tell from the data sheet if both ends are connected, but I have to guess that one end is still open...it's just soldered down in order to hold it in place.  If you bring this arrangement too close to the ground plane, it will add capacitance to the circuit and detune it a lower frequency.  Some of the energy will also be coupled to the ground and be lost, or at least upset the intended radiation pattern.