That's the old chestnut - how many weapons does
Khan's ship have?
Apart from the use of the Genesis device as a weapon, the movie only shows two types of weapons: torpedo tubes (easily established as Kirk 2, Khan 4, even though Kirk has something called "Torp Bay 4"), and phasers. There is no third type of weapon in evidence: we don't hear of any "phaser cannon", nor do we see one in action.
Kirk's ship has eighteen phaser turrets: three twin ones above and three twin ones below the saucer, four single ones below the secondary hull, and two single ones atop the shuttlebay. Khan's ship has either sixteen or eighteen phaser turrets: three twin ones above and three twin ones below the saucer, possibly two single ones next to the impulse engine (visible in some backstage photographs of the model, not visible in the later incarnations of the ship in TNG or DS9), and then two single ones on the outer sides of the cylinders on the "roll bar".
Kirk's ship only ever fires from the twin banks below the saucer. Khan's ship only ever fires from the single banks on the sides of the cylinders. Both ships shoot identical beams, with identical results against unshielded ships.
There are no beams coming from the cylinders themselves, that is, from their forward or aft ends.
The beams Khan first fires are drawn as beginning from the sides of the cylinders, where these single yellow-painted ball turrets exist. There is a white light coming from the forward end of the cylinder, function unknown.
If the white light thing is a weapon, we are left wondering why Khan never used it. If some of Khan's phaser turrets are more powerful than the single ones on the sides of the cylinders, we are left wondering why Khan never used them. In the end, it seems Khan only had one type of phaser weapon available, and all sixteen or eighteen of them had equal destructive power, so the choice of one over another was arbitrary.
Naturally, we may invent reasons for Khan's "odd decisions". Perhaps he never learned to use the white-light weapons? Perhaps using them would have called for more crew than he had? But we may just as well decide that the white-light things were navigational deflectors or sensor pods or communications arrays or just plain searchlights. They did not affect the balance of power in ST2:TWoK, and they never saw action in any other incarnation of Star Trek.
FWIW, the movie does not suggest that Khan's ship would be newer than Kirk's, or more advanced. She has the same interiors, the same controls, even the same color scheme. Her registry is higher than that of Kirk's ship, but not by much, and we already saw registries that high back in TOS, suggesting that both the hero ship and the villain ship underwent a refit to their movie specs after serving in a TOS style guise previously...