I imagine it would depend on the adaptor. If it’s an “active” adaptor (ie, there is some processing done on the signal using a microprocessor), that introduces at least some lag.

A “passive” adaptor or a (passive) cable should have no latency otherwise.

In terms of quality, there shouldn’t be much difference if any, though it may depend on the particular adaptor you use.

My adapter and most I found could not do 120Hz at 4K, but the cables were easier and cheaper to buy.

In my experience, in IT, cables are usually the way to go over adapters. Adapters tend to break more often than cables.

Someone else may offer a more thorough differing opinion… like perhaps a very high quality HDMI cable with a high quality adapter may be better than a mediocre off-brand standalone cable, but if the prices/brand are the same… I’d stick with the single cable.

Before USB-C became standard, we stuck with HDMI cables and then had different adapters on hand just for convenience, but if we knew a cable is going to be used in the same equipment for a long time, we’ll try for the single cable.

I just know I couldn’t use any of those to get 4k. I needed to get a cheap ATI Radeon card with direct HDMI output and a high quality HDMI to HDMI cable to get 4k. I had to make sure both support 4k. The card and the display will do a very short test of the cable on each connection and if it doesn’t carry enough bandwidth and fails, the card will tell the OS and 4k won’t even show up for selection.

Edit: from the other answers, I’m probably wrong - maybe don’t trust this as correct

I don’t think so - HDMI and Display Port actually carry their signals in the same way, so the adapter is basically just converting between two plug types without any smarts in the middle.

In theory you could get an adapter that is badly made and adds some noise to the signal or something and forces the monitor to down-spec it’s signal but I’m not sure how likely that is to come across.