There are probably a variety of factors. How fast they form, how fast they sink through the atmosphere. Like rain, there's likely a part in the atmosphere where they form and grow, and a part, deeper in the atmosphere where the temperature gets too high and the diamonds dissolve again into plasma, perhaps releasing some free/charged carbon to return upwards and be part of the cycle all over again.
If I was to guess, I would guess that the diamonds formed in gas giant atmospheres would be much more impure. The interior of the Earth is relatively slow moving and consistent in temperature, often cooling very slowly. It's the same process which helps form veins of metals form, which are followed in mining. This very gradual cooling in Earth's core probably plays a role in the purity of diamonds. Earth's core also has very little nitrogen, Diamonds formed in Earth are about 99.95% carbon atoms on average, with nitrogen, often the primary impurity.
The relative abundance of nitrogen available in gas giant planets suggests to me that on average, gas giant diamonds would be more impure. Less consistency in temperature during formation and an availability of nitrogen suggests to me that 99.95% pure carbon diamonds are unlikely. There's also the problem of how large they'd be. If diamonds form slowly, which seems likely and as they grow larger, they would have a density problem, likely slowly falling deeper into the gas giant's atmosphere, where they would get too hot and likely become plasma and dissipate.
If I was to guess, I'd guess that the diamonds that form inside the 4 gas giant planets in our solar-system are more along the grain of sand to perhaps raindrop size and probably quite impure. But I'm mostly guessing. There might be a planet out there somewhere, (perhaps a mini Neptune) with just the right mix of lower gravity and possible suspension and temperature where it forms grapefruit or basketball sized diamonds of high purity. It's certainly possible, but my guess is that diamonds inside our solar-systems 4 gas giants would be pretty uninteresting, though the chemical and gas-giant geological processes that form them would still be interesting, but I don't think the diamonds would stand out as gem quality or worth harvesting.