Quick and dirty answer is that the AMD 5xxx and 6xxx series CPU's generally run cooler in otherwise identical laptops vs intel 10/11/12th CPU models (12th gen especially seem to perform very well but run quite toasty).
Overall performance is the same and imagine noise levels will be similar when either laptop is under full gaming load. Generally modern gaming laptops have multiple shared heatpipes between the CPU and GPU in addition to some form of power shifting technology - AMD Powershift or Nvidia Dynamic boost. Essentially this means they will keep boosting right up to thermal thresholds thus running relatively loud (compared to say a desktop).
There are exceptions of course. The Alienware X17 series are generally considered extremely good when it comes to noise levels despite having top tier components. Downside is they cost an absolute fortune.
I will say my Legion 7 with its Ryzen 5800H and RTX3080 seems to do an excellent job maintaining reasonable noise levels and keeping temps in check. Spent around 3 hours last night in Destiny 2 and despite the GPU running at ~150-160w constantly it remained pegged at 70c. CPU was nudging into the high 70's / low 80's but even then it was quite happy at 4-4.2Ghz across 4-6 cores. The vapour chamber cooler is obviously doing the work. I should note that I run a "custom" cooling pad whenever I do anything "heavy" on the laptop. It is a Cooler Master U2 with 2x Corsair ML120mm fans running at half speed through a 5v USB to dual fan splitter.
The latter point is something I would recommend regardless of laptop choice. The extra cooling won't make a huge difference to temperatures but in my experience it means the laptop will run its fans at a notch or two lower thus making it quieter overall (its the fan pitch which is most noticeable).
In terms of the Omen range I can honestly say I am quite impressed. We have a few of them at work being used for relatively heavy CAD work and whilst they are not as "quiet" as my Legion they don't wail like a banshee either. All of them have Ryzen 5800H CPU's and a mixture of RTX 3070 and RX 6600M GPU's. Overall build quality is excellent with good keyboards, bright clear screens (1440 165Hz models) and a good selection of ports. Biggest feature for us was the USB-C port having DisplayPort alt mode as they are used with a dock (USB-c charging relatively pointless as the laptop needs more than ~100w the cable can supply). Major bonus points as well in that they are easy to get into and upgrade, something which I can't say is true for my Legion - sodding RGB strip gets in the way.
Another option is the Legion 5 and Legion 5 Pro which are often highly regarded for the price point.
Ultimately it all depends on what you want to spend and what your expectations are. If you just want to get the best bang for buck then look at something Ryzen 5800H based with at least a RTX3060 or RX6600M. Should be be quite do-able in the region of £900-1,000 depending on deals (I think our Omen 16's with the 6600M were ~£940).
TL
R:
Ryzen run cooler than Inter & perform generally the same outside of 12th gen when you start ramping up the power
Overall noise levels are more a product of the laptop cooling design & power shifting tech rather than the components installed (exceptions of course)
Buy what suits your requirements best at the budget you want to spend. No point getting a Ryzen based machine if you want Thunderbolt 4.0 for example.
