What benefits does Automatic Dependent Surveillance bring to continental ATC over Mode S?
ADS-B is more accurate.  SSR calculates the distance to a target based on when it receives a reply from the transponder, which is inherently imprecise, and calculates the azimuth by the sweep, which is also inherently imprecise.  A blip on a radar scope isn't exactly where the plane is but rather the center of an area where the plane is most likely to be.  ADS-B means ATC knows exactly where the plane is, which means they can (eventually) use tighter spacing if needed.

ADS-B is more reliable.  If two aircraft are near each other, their transponder replies may overlap, making both unreadable--and at the time it is most critical for ATC to be able to see both aircraft for separation.  ADS-B messages are transmitted randomly, which does result in occasional errors but means that two aircraft near each other aren't any more likely to step on each other than two far away, which is a substantial reduction in risk.

ADS-B can improve TCAS by not requiring air-to-air interrogations, reducing the total number of mod eS messages being broadcast and therefore the odds of aircraft stepping on each other.

ADS-B updates faster.  SSR relies on a radar slowly sweeping around, which means targets are only updated every 5-10 seconds, and this magnifies the effect of message errors.  This is "solved" in busy terminal areas by having several overlapping SSRs, but that also increases the error rate so it's of limited use.  ADS-B transmits position randomly every second, and even if some overlap, the readable update rate is still far higher than SSR.

ADS-B is cheaper.  You can put simple ADS-B receivers in many places where expensive SSR wouldn't be economically justified, giving ATC much improved coverage, particularly close to the ground and in mountainous areas.  You can also reduce SSR overlap, even if SSR doesn't go away entirely, which also saves money.

ADS-B provides improved situational awareness to pilots.  SSR doesn't provide any traffic data to pilots, and even TCAS isn't all that accurate (which means lots of false alarms) despite being extremely expensive.  ADS-B cheaply shows pilots exactly where the other traffic around them is--even non-participating traffic in an area with TIS-B.

Future ADS-B enhancements will also show aircraft intent.  SSR (and current ADS-B, to be fair) only show a plane's current position, often with a line projecting where it will be in N seconds/minutes if it continues along the same track since the last position.  ADS-B allows an aircraft to indicate what it will do in the (near) future, which is very useful when you have two aircraft near each other and one or both are turning and thus the straight-line projections aren't accurate.

The inaccurate flight paths you see on FR24 et al are due to aircraft not having ADS-B; the jagged tracks are generally from mode S-only aircraft, and the receivers have software that uses multilateration (MLAT) to cooperatively estimate where non-ADSB messages came from.  The FAA has experimented with this tech themselves in mountainous areas, and it's better than having nothing at all, but it's still far less accurate than SSR, much less ADS-B.