The vent line is to release any vapors from the pump back to the suction source.

The recirculation line is usually installed to allow the pump to maintain a reasonably high flow rate even when much of it isn't required at the pumping destination. Under these conditions the line to the destination is closed (or severely throttled) and the recirculation line is opened to permit the flow back to the suction source. This allows the pump to continue to operate at a flow rate close to the best efficiency point, and thus operate efficiently and reliably.

Please note that the recirculation should never be taken back to the suction inlet to the pump as it will cause excessive turbulence and the resultant air entrainment problems.