Intel's been having problems for the last couple of generations with cpus burning out, so if you want new, I'd pick AMD. They're also winning on the perf and price fronts for now. The previous gen tends to be cheaper than the current one if you don't need latest perf.
Tower/mini depends on personal preference really, whether you need optical drives, many disks, etc.
As someone else has mentioned, it depends on your use case.
I'm running Tiny Core almost exclusively on laptops, mostly HP, none of them newer than 2018. Some Intel, some AMD. They tend to be quieter and less power-hungry than desktop/tower boxes. They all handle Tiny Core beautifully (even the ones with touch screens!) though it's often an adventure trying to get wifi working.
My backup server / file dump is a 2012 Dell Optiplex 790 and it's fine - though probably soon to be replaced with a 2018 Lenovo Thinkstation E31 (that only draws about 22 watts (currently powered off, so 0 watt)) Those are both Intel based but, like the laptops, old enough that current Intel vs AMD caveats probably don't apply.
If you're looking to buy something new, let your use case and your budget be the determining factors.
If you're looking for used, I heard somewhere that the late-1900's vintage Deskpro EN boxes were pretty durable.