Not for this monitor/graphics card combo. Any ten-year-old port/cable should be fine.
The differences are only apparent in higher-end monitors, like the 4k resolution, 240 Hz, ultra-mega-super-duper-wide ones (something like this: 49” Odyssey Neo G95NA DQHD LED 144Hz 1ms(GtG) Curved Gaming Monitor - LS49AG952NNXZA | Samsung US). In those cases, you either need a very recent version of HDMI + nice cables that support the full bandwidth, or it can generally be safer to use DisplayPort.
But that’s more a matter of some older versions of HDMI and some inferior cables not having enough bandwidth to support that combination of resolution and refresh rate.
But in your specific case, 1920x1080 @ 100 Hz will work fine with either. There will be no difference in image quality.
The connector ports and cables just determine the maximum refresh rate and resolution you can use (and whether you send audio over them, which you shouldn’t do with computer monitors anyway because they sound like crap). But if it’s within the spec, it’s a digital signal, so “better” cables or connectors won’t matter and won’t affect the image quality.
How it looks on your monitor is more about that monitor’s LCD panel type (IPS vs VA and others), which have various tradeoffs between color gamut (what they can display), accuracy, and various visual artifacts. OLED monitors will give you very good vibrancy with some other downsides.
It’s all too much to get into here, but you can look into those if you really care. Probably you don’t if you’re hunting for budget monitors anyway
It also kinda depends on how well calibrated for color and gamma your monitor is. If you’re not sure, find a pretty game screenshot that you like, try to remember how it looks, and then go to an Apple store and pull that same screenshot up on one of their Macbook Pros. Those have really nice panels with good color gamut and accuracy and are calibrated ahead of time, so you can see how the image is “supposed” to look when everything is working. Then you can decide if it’s worth the price difference (between a good and bad monitor, not a Mac). For many people it’s not even noticeable, like the hordes of poorly-calibrated HDTVs you’ll find at every Costco, Best Buy, Target, etc.
Beyond that, if you want good gaming graphics but only on occasion, I suggest using GeForce Now instead. That gives you an RTX 4080 in the cloud, with full support for ray-tracing and DLSS frame generation. It will look way way way better in modern games than a 2080. They’re probably going to upgrade to 5080s soon too, once those come out.
They have both monthly plans and day passes available, and for $20/mo it’s a no-brainer vs having to upgrade your expensive graphics card every few years.