What are the differences and benefits between/of time-based vs distance-based separation?
There is more information available from NATS on the website on Time Based Separation [TBS] and from a very descriptive Infographic on TBS PDF. While Heathrow (EGLL) is the first airport where this is being researched and tested, there are als other works, such as a research paper called Potential Benefits of a Time-based Separation Procedure to maintain the
Arrival Capacity of an Airport in strong head-wind conditions PDF which applies TBS at Charles Des Gaulles (LFPG). 

The main focus of using TBS is to increase airport capacity during strong headwind operations, which significantly decrease groundspeed and therefore traffic flow. 

None of the papers I have found speak of hazards related to decreasing the lateral distance between flights. The general idea seems to be that the minimal safety gain accompanying lower groundspeeds due to strong headwinds (lower GS means lower closure rate to objects / stationary traffic on ground) can be partially sacrificed and offset by TBS to maintain airport capacity or reduce the impact of the strong headwinds on airport capacity. This sacrifice becomes available since stronger headwinds also aid the dissipation of wake vortex.

The below table is based on own calculations, assuming constant approach speeds.
It should show you how the time to cover a set distance increases with decreasing ground speeds, which means that e.g. it takes longer to cover 3nm to reach the runway threshold. Reducing the lateral separation (the values chosen are deliberately a bit extreme, as none of the papers mentioned a formula to calculate TBS distances) reduces time to cover the distance, but covering 2.5nm at 130 GS is becoming quite close to 3nm at 150 GS.

+-----+-----+-------+-----+
| GS  | 3nm | 2.5nm | 2nm |
+-----+-----+-------+-----+
| 150 |  72 |    60 |  48 |
| 140 |  76 |    64 |  51 |
| 130 |  83 |    69 |  55 |
+-----+-----+-------+-----+