The verdict

CPU
ASUS NUC 13 Pro i7 has 1.3x the CPU throughput (21,000 vs 16,372 PassMark).
Plex
ASUS NUC 13 Pro i7 transcodes more 4K HDR streams (~8 vs ~5).
Idle power
HP Elite Mini 800 G9 idles lower (5W vs 9W) - cheaper to run 24/7.

Which wins for each job

Job ASUS NUC 13 Pro i7 HP Elite Mini 800 G9
Plex / Jellyfin ~8 streams win ~5 streams
Frigate NVR ~6 cameras win ~4 cameras
Proxmox / VMs 21,000 PassMark win 16,372 PassMark
Firewall / router 1 NICs 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 ASUS NUC 13 Pro i7 HP Elite Mini 800 G9
Price (list) $599 $550
CPU Intel Core i7-1360P Intel Core i5-12500T
Cores / threads 12 6
PassMark 21,000 16,372
iGPU Iris Xe UHD 770
NICs 1 1
NIC model Intel I226-LM Intel I219-LM
RAM max (GB) 64 64
M.2 free 1 1
PCIe slot none none
Idle (W) 9 5
TDP (W) 28 35