The verdict

CPU
HP EliteDesk 800 G6 Mini has 1.9x the CPU throughput (10,221 vs 5,500 PassMark).
Plex
Beelink EQ12 transcodes more 4K HDR streams (~5 vs ~3).
Idle power
Beelink EQ12 idles lower (7W vs 11W) - cheaper to run 24/7.
Networking
Beelink EQ12 has more NICs (2 vs 1) - better for a firewall or VLAN trunk.
RAM ceiling
HP EliteDesk 800 G6 Mini takes more RAM (64GB vs 32GB).

Which wins for each job

Job Beelink EQ12 HP EliteDesk 800 G6 Mini
Plex / Jellyfin ~5 streams win ~3 streams
Frigate NVR ~4 cameras ~4 cameras
Proxmox / VMs 5,500 PassMark 10,221 PassMark win
Firewall / router 2 NICs win 1 NICs
Local AI / LLM no slot no slot

Approximate - capacity estimates from curated specs, the same logic as the recommender. Verify against your exact setup. Prices move; check the retailer.

Full specs

Spec Beelink EQ12 HP EliteDesk 800 G6 Mini
Price (list) $189 $185
CPU Intel N100 Intel Core i5-10500T
Cores / threads 4 6
PassMark 5,500 10,221
iGPU UHD UHD 630
NICs 2 1
NIC model Intel I226-V Intel I219-LM
RAM max (GB) 32 64
M.2 free 1 1
PCIe slot none none
Idle (W) 7 11
TDP (W) 6 35