Short answer
Hire the A7 IV if your shoot is more than 50% stills, if you need to crop heavily, or if you're shooting weddings, portraits and lifestyle. Hire the A7S III if your shoot is mostly video, you're working in low light, or you need 4K120 slow motion.
Stills
The A7 IV's 33MP sensor lets you crop into a frame for a tighter portrait, deliver large prints, and still keep enough resolution for double-page magazine spreads. The A7S III's 12MP is fine for screen and 4K stills, but it'll let you down for print at any size.
Dynamic range is comparable in good light. In bad light, the A7S III pulls ahead — usable shots at ISO 12,800 with very little noise.
Video
Both shoot beautiful 4K. The A7S III shoots 4K120 internally with 10-bit S-Log3, which the A7 IV cannot do (it tops out at 4K60 in S35 crop). For wedding film, music videos, or anything where you want slow-motion B-roll, the A7S III is the obvious pick.
For straightforward 4K30 or 4K60, the A7 IV is more than enough and you keep the option of high-res stills.
Low light
The A7S III's dual native ISO (640 and 12,800) is genuinely a generation ahead. Reception venues, dawn surfs, indoor weddings without flash — this is the camera. ISO 12,800 looks like ISO 800 on most cameras.
Autofocus
Both use Sony's latest real-time tracking and Eye AF, and both are excellent. Practical difference is negligible — the A7S III edges it slightly in very low light because the sensor sees more.
What we'd hire for common shoots
Wedding: A7S III as A-cam (low-light reception + slow-mo) + A7 IV as B-cam for stills.
Travel content: A7 IV. One body, does everything, lighter.
Music video: A7S III. Slow-mo and clean low light wins.
Real estate: A7 IV with the 16-35 Zeiss. Resolution matters.
Surf at dawn: A7S III with the 70-200. Pulls usable colour at -1EV.