The training response you get from barbells, dumbbells and resistance bands are all a bit different. They can each be used for an effective workout independently, or combined in various ways to enhance your workout.
With the barbell, you don't need to use as many stabilizers as with dumbbells but you can generally go heavier.
With dumbbells, you can't necessarily go quite as heavy, but you have to stabilize the weight with each arm independently, so you hit the muscles differently.
With resistance bands, the resistance progressively increases as the band stretches, allowing you to emphasize heavy loads near lock-out. They also load the joints differently than barbells or dumbbells because you're not lifting or holding up physical weight. For instance, using a resistance band for a chopping or twisting motion allow you to load your muscles and joints heavier relative to the lateral movement pattern without the vertical loading involved with holding up a physical weight. That's not necessarily better or worse, just different... and can be more or less applicable depending on the individual's needs. Resistance bands can also put less strain on joints due to the lower physical mass involved in the movements.
The barbell, dumbbell and resistance bands are just some of the tools available for strength training. What you use and how you use it should be primarily determined by what's easily available to you combined with your personal goals.