The cytoplasm of both cells has a higher osmolarity/ lower water potential than the distilled water, this causes water to enter the cells via osmosis along the water gradient. The red blood cell is surrounded by a plasma membrane which is relatively weak and bursts with the increased pressure due to the water, whereas the plant cell also contains a cell wall. The cell wall provides more support and so the plant cell wouldn't burst but become turgid, as turgidity increases water influx decreases.