Does the HDMI cable make a difference in Apple TV quality?

Please pay attention to the bandwidth (Gbps) that HDMI cables are rated for (5-10-18-48). That is the most important cable property. Roughly speaking, if the content potentially generates more data than the cable or ports can handle, then the sending device will throttle down on quality: resolution, frame rate, chroma subsampling, dynamic range. It will indicate maximum video settings at a lower level.

[Ignore the ethernet support in some HDMI cables, as no consumer set-top box, TV, receiver, or soundbar supports that. HDMI Ethernet Channel functionality is only available if both linked devices are HDMI Ethernet Channel-enabled.]

HDMI cables - Different cable types - HDMI.org

The used cable should at least match the HDMI standard of either one of the HDMI ports to which it connects. Else the cable itself might be the bottleneck for data throughput.

1. If one of you devices to which the cable connects has an HDMI 1.4 or lower port, then a 10 Gbps HDMI cable is all you need. Higher rated cables wouldn’t perform better in this setup.
2. If one of your devices to which the cable connects has an HDMI 2.0 port, then an 18 Gbps HDMI cable is what you need. Higher rated cables wouldn’t perform better in this setup.
3. If both devices to which the cable connects have HDMI 2.1 ports, then a 48 Gbps HDMI cable would bring support for all the HDMI 2.1 features.
4. If you are going to buy a new cable anyway, then you can of course buy a better-than-strictly-needed cable now to future-proof yourself, for when you get devices with HDMI ports at a higher HDMI standard.

You can search out the details, but HDMI cables conform to various iterations of the HDMI standard. As long as the cable meets/supports the necessary HDMI revision level, you're good. Longer lengths, say, over 10', may benefit from more expensive construction and better sheilding.

I reject assertions that expensive golden monster ultra-marketed cables deliver a better picture or sound than a cheap cable that meets the spec and has passed a reasonable QA test. Of course, folks are welcome to psych themselves into thinking their expensive HDMI cable purchases were worth it.

Length, yes. 1m (3 feet) or less is best.