From this column, the third entry down the page…
[QUOTE=GGlatfel, via the Internet]
Since light has a particle nature, and since photons have mass, and since our sun has been shining for 15 billion years, why aren’t we knee-deep in photons?
[/QUOTE]
We aren’t knee-deep in photons because photons are often absorbed.
We aren’t knee-deep in neutrinos because neutrinos are rarely absorbed.
Which is why you can feast on them over the holidays and not put on the pounds!
Hopefully one of our scientific specialists can provide a definitve answer,
but I believe part of the solution is that photons do *not have actual *mass,
but rather a property known as “rest mass”. Photons are usually in motion,
hence are massless most if not all the time.
Is rest mass merely theoretical, or can you actually stop a fundamental particle? Once stopped, how do you start it up again?
Other way around! photons have mass (and momentum) – if they bump into you, you get bumped – but they have no “rest mass”: when they come to rest, there is no mass left.
The mass of a photon is given by the famous equation: e=mc squared
Or, re-aranging, m = e/cc
Energy of a photon is hfreq (UV is more energetic than IR because the frequency is higher), so mass = hf/cc, h is Plancks constant, and the result is a very small number indeed.
However, it has been demonstrated by experiments such as balancing a small weight on a beam of light.
Photons have energy and momentum, but a photon does not have mass.
E = m c^2 is only correct in the limit of no momentum. Including momentum p, you have instead:
E^2 = m^2 c^4 + p^2 c^2
Who ever said we are not up to our knees in photons. They are everywhere I look.
Neutrinos rarely get stopped, usually they keep going, and the universe is a BIG place.
Photons get stopped a lot more easily, and can bounce around, but they also get absorbed, sometimes in eyes to trigger sight, often as heat.
There’s that, too. 
Good equation for entry-level physics, but not helpful when explaining the universe to my mother.
It’s got momentum, it’s got weight: the effect is that it has mass. If it looks like mass, and smells like mass, and tastes like mass, why not call it mass?
The equation you give provides an answer: when considering items that DO have rest mass, that is, items which are not photons, you can’t add the two kinds of mass together.
For photons, your equation comes out to the same result:
p = mv = mc, so
E^2 = (rest mass)^2 c^4 + (non-rest mass)^2 c^4
Since the (rest mass) of a photon is zero,
E^2 = (non-rest mass)^2 c^4
or
E = m c^2
where m is representing the non-rest-mass