Sound wise, the R4 probably has better insulation. Airflow wise I don't think you'd have issues in the P180 Mini in idle, if you can fit the GTX 670 in there. The R4 has the side panel hole available, but that's just another hole for noise to escape through if you pop it open. I'm not convinced the bottom intake in the R4 is any more beneficial than opening the P180 Mini's compartment hatch.
Obvious improvements before swapping cases would be getting quieter (more CFM/dB) fans, removing/isolating HDDs and declining the 670 in favour of a cooler-running card. The i7-3770K makes me think you're over-investing here.
Case: Define Mini
Parts: P8Z77-M Pro µATX, i5-3570K, STRIX-GTX960-DC2OC-4GD5, 8 GB G.Skill DDR3U, Xonar DX, WD G 1 TB, mx100 256 GB SSD, RX-5300 PSU
Cooling: Noctua NH-U12P SE2 + Scythe SS PWM, 2x Noctua NF-P12
Extras: Eaton UPS, Dell 24" EIPS, CM QuickFire TK, Logitech G303, Synology DS213j 6 TB NAS