The verdict

CPU
HP EliteDesk 805 G8 Mini has 1.2x the CPU throughput (21,064 vs 18,159 PassMark).
Plex
Beelink EQR6 transcodes more 4K HDR streams (~5 vs ~3).
Idle power
Beelink EQR6 idles lower (10W vs 13W) - cheaper to run 24/7.
Networking
Beelink EQR6 has more NICs (2 vs 1) - better for a firewall or VLAN trunk.

Which wins for each job

Job Beelink EQR6 HP EliteDesk 805 G8 Mini
Plex / Jellyfin ~5 streams win ~3 streams
Frigate NVR add a detector add a detector
Proxmox / VMs 18,159 PassMark 21,064 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 EQR6 HP EliteDesk 805 G8 Mini
Price (list) $279 $280
CPU AMD Ryzen 5 6600H AMD Ryzen 7 PRO 5750GE
Cores / threads 6 8
PassMark 18,159 21,064
iGPU Radeon 660M (RDNA2) Radeon Vega 8
NICs 2 1
NIC model Realtek RTL8111 Realtek RTL8111
RAM max (GB) 64 64
M.2 free 1 0
PCIe slot none none
Idle (W) 10 13
TDP (W) 45 35