Since this is a very common question I thought I'd post it here. This is direct from the Yamaha website.
The RX-V677 has the following differences/advantages over the RX-V675:
1. Latest HDMI features: 4K video
and 18 gbs bandwidth. (RX-V675 had 10.2 gbs.)
2. Apple Lossless audio playback
3. Gapless playback compatibility allows play without interruption
4. Wi-Fi Built-in with Wireless Direct. (No router needed for AirPlay or Play to Music.)
5. HTC Connect™ for music streaming from HTC Connect™ compatible smartphones
6. Background Video feature for Radio/USB/NET/Audio inputs. RX-V675 has Tuner and Net sources only.
7. Extra Bass enriches powerful bass sound, even with small speakers
8. Virtual CINEMA FRONT provides virtual surround sound with 5 speakers in front.
9. YPAO Volume(adaptive EQ) Maintains balance of bass, mids and highs regardless of volume level.
10. Allows for DSP changes from the AV Controller. (App 4.0 and RX-Vx77 models only.)
11. Upgraded GUI. (Better icons and look/feel.)
12. RX-V677 does not have a Preset programable remote
RX-V677 17-1/8” x 6-3/4” x 14-3/8" (with antenna: 17-1/8” x 9-1/4” x 14-7/8")
RX-V675 17 1/8” x 6 3/4” x 14 3/8”
Power and Weight spec's are the same:
105w x 7 Rated Output Power (8 ohms, 20 Hz-20 kHz, 0.09 % THD, 2 ch driven)
90w x 2 Rated Output Power (8 ohms, 1 kHz, 0.9 % THD, 2 ch driven)