It is possible to lose weight with diet alone, however, exercise burns calories and spikes your metabolism, which also burns calories and should be your main calorie burning furnace. Losing weight and/or adding muscle is roughly 30% exercise and 70% diet and rest. In other words, diet > exercise.
Also, resistance training/weight lifting for 60 minutes has been proven to burn more calories than 60 minutes of running at a 7 minute mile pace. Resistance training/weight lifting adds lean muscle, which burns fat. It's almost like turning the fat into muscle. Hope this helped!