I've been told that EM beams are "gaussian" beams that spread nonlinearly, so the spot size grows faster and faster as you travel further away.
However, when you talk about an antenna, it is defined by a beamwidth expressed in degrees, which makes it seem like there is a constant growth of the spotsize.
So do antennas not output gaussian beams, or does the beamwidth express something different than I am thinking?
(in addition to ThePhotons succinct answer)
Antenna can have many beams depending on shape, Lambertian response from Pad Antenna like SMD LEDs, or Torroidal response from Dipoles. Gaussian is not a common shape for EM Antenna but very common for light sources. A few exceptions perhaps similar are very high gain Helix or high precision parabolic dishes but not perfect Gaussian shape.
Gaussian shapes are more likely to define light beams such as some LED's or Lasers with a Gaussian lens.
e.g. 
Full Width Half Maximum, FWHM , is equivalent to -3dB power or 50% Iv peak.
where full angle \$\Theta=2\theta\$ of half angle
Above are 3 different hypothetical Gaussian profiles.
The offset one is not as bad as above but common for IR LED's which is why Half Beamwidth is used for these device specs and Full Beamwidth is used for visible LEDs.
This is because IR LED parabolic lenses are very small and chip angle may be offset. Visible LED's use transparent substrate with parabolic reflectors and only a few are Gaussian in profile. ( although some are very smooth Gaussian-like)
Perhaps some IR LED OEM's are using Full angle now to avoid this confusion.
Here is a smooth Gaussian laser response.
5mw Green laser
For advanced principles on Gaussian Light http://marketplace.idexop.com/store/SupportDocuments/Gaussian_Beam_PropagationWEB.
You can express the solution to the electromagnetic wave equation using various different basis functions. Usually we choose a basis set that is convenient for matching up some boundary conditions that apply to our problem. The gaussian beams are a convenient basis set when working with a beam emitted from an aperture. But you might prefer the spherical harmonics if your antenna radiates more isotropically, for example.
TLDR: Gaussian beams are just one choice of how to express the emission from an antenna.
The gaussian beams do diverge with an angle, which for the 0th order beam is given by $$\Theta = \frac{\lambda}{\pi w_0}$$ where \$\Theta\$ is the full angle of divergence (not the half-angle), and \$w_0\$ is the beam's waist diameter. So there's no contradiction between saying that a radiation pattern is a gaussian beam and saying that it diverges with a known angle.
The shape of antenna output also depends on the type of antenna. In the case of an omnidirectional antenna, electromagnetic radiation is uniform in all the direction and shape is spherical in nature. While in the case of Directional antenna ( like antenna arrays), electromagnetic radiation or energy is more in a particular direction compare to other direction so the shape is like a pencil beam. There are many other specific shapes generated by shaped beam pattern antennas.
