This is a good question.

First, one needs to make a choice of 3+1 dimensional quantum field theory in which a massless spin-2 graviton obeying the Bose-Einstein statistics appears as the carrier of the gravitational radiation from (classical) General Relativity. There are gravitons in higher-dimensional theories, string-theoretical and otherwise, and there are also 3+1 theories with massive gravitons and/or different spin statistics.

So below I'll be talking about gravitons in perturbative quantum gravity and canonical quantum gravity, two specific quantum gravity theories (i.e., "two QGs").

If we make such a limitation, then there are some noteworthy differences between massless spin-1 photon and massless spin-2 bosons: universality, coupling, spin, and background. While these make these boson fields not the same, it does make them pretty comparable, and allows for successful analogizing.

We'll start with universality: not everything feels electromagnetism, so the photon couples non-universally. In particular it does not couple to neutrinos, and does not self-interact (in the absence of charged matter that feels electromagnetism). Gravitons interact with everything, including neutrinos and photons, and including other gravitons (even in the absence of any other matter).

"Matter" in the paragraph above is tricky - the photon does not typically appear in isolation in a quantum theory. In quantum electrodynamics and the Standard Model it has at least an electron/positron field where its partner charged particles can be found. Since in the two QGs a lone graviton is one of a large number found in gravitational waves in classical General Relativity, and since the latter admits vacuum solutions with gravitational waves, we conceptually 'promote' gravitons into matter, even though the stress-energy tensor of the (classical) Einstein Field Equations is set to zero in a vacuum solution. As such, gravitons themselves are gravitationally charged. Universality of free fall means that all other particles -- photons, neutrinos, ... -- also are gravitationally charged.

When a boson interacts with its appropriately charged matter there is a "coupling", which can be constant, or which can depend on energies ("running coupling" or "effective coupling"). As energies increase, both photons and gravitons depart from their default couplings, the fine structure constant \alpha and Newton's constant G. Perturbative methods for the running coupling of photons are better known (the so-called "beta" function of QED for instance <https://en.wikipedia.org/wiki/Beta_function_(physics)#Quantu...>), but the failures of a perturbative running coupling for gravitons is more famous: that's where the "gravity is non-renormalizable" comes from.

For these two types of bosons, the spin determines what happens to charged matter exchanging them. For spin-1 photons, two similarly-charged particles repel and two oppositely-charged particles attract. For spin-2 bosons either (i) there is only one charge, and it's always attractive or (ii) there are two charges that differ by sign, and similar charges attract but opposite charges repel. There is no evidence for the (ii) option, although there are plausible reasons why we might not have found any. The coupling strengths of gravitons are very weak compared to photons, but as with photons masslessness means infinite range. It could be that all matter with the opposite gravitational charge (gravitons, electrons, ..., maybe with some supersymmetry-like partners, or other weirder particles that aren't much like the Standard Model ones) have been pushed out of our observable universe through "anti-gravitation", and there is no decay path to such matter that exists in our region. Who knows. Option (i) is completely consistent with evidence and simpler.

Finally, background: General Relativity has a "no prior geometry" principle. This means that moving matter generates spacetime curvature. If matter moves differently, it generates different curvature. Following the slogan, "matter tells spacetime how to curve, curvature tells matter how to move", there is one collection of moving matter for each different dynamical ("unfixed") spacetime. Perturbative methods like the two QGs above use a fixed background curvature; the Standard Model and QED effectively do this too. There are then arbitrary numbers of distributions of matter that are associated with the background spacetime, and we then have to add extra gravitational information to account for the energy-momentum of those distributions. That information is typically a gravitational backreaction. In other words, when we chop a dynamical spacetime into static flat spacetime and a dynamical component, we have more bookkeeping to do.

One can turn this around a bit. In General Relativity nobody's timeline is precluded from calculating the whole spacetime. It's easier for some timelines, it's harder for others. But one always has choice. In relativistic QFTs one picks out a universal, absolute timeline against which eigenvalues evolve. One thus wants to chop up a General Relativistic spacetime into spaces organized on one timeline (arbitrarily chosen out of infinite options), and then introduce e.g. the "lapse" and "shift" functions from canonical quantum gravity. These functions do not represent anything physical. These functions are needed to deal with the fact that we have approximated a dynamical spacetime with a fixed time along which we arrange successive spaces. Especially when one also carves each space into a fixed-background and dynamical part as in the previous paragraph, one introduces approximation artifacts and loses high-frequency information. For the same spacetime with the same matter moving identically, we can get quite different lapse/shift functions (or bookkeeping fields) if we switch from spaces arranged on one timeline to spaces arranged on a different timeline.

QED and the Standard Model basically ignore these issues: they are theories defined against static flat spacetime. The bright side is that General Relativity guarantees that at every point in any spacetime (no matter how strong the curvature is) there is always a small (and that may mean microscopic or ultramicroscopic or ...) patch which is flat. It's like zooming in on a picture of a circle: zoom in enough and you lose sight of the curvature. It's also like blowing up a circle's radius larger and larger. Humans see this all the time standing on the ground and not noticing the curvature of the Earth's surface.
Make gravity strong and the radius of curvature shrinks; make the moving matter gravitationally relevant and the "circle surface" gets all jagged and bubbly. Relativistic quantum field theories like QED and the Standard Model can be adapted reasonably well when gravitation is weak and when appreciable sources of gravitation move very slowly compared to the speed of light.

Finally, in spite of these differences one can draw some analogies with different levels of formality and with different domains of applicability between photons (and electromagnetism) and gravitons (and gravitation). One generally does this classically though, because of the large masses (and thus particle numbers) involved, so the analogy is really between the Maxwell-Einstein equations and e.g. Gravitoelectromagnetism (GEM). See <https://en.wikipedia.org/wiki/Gravitoelectromagnetism> if you're curious.