This is a very popular topic amongst saxophonists. Let me start out by saying, there is no right answer.
Cane reeds have been the mainstay for years and the most 'traditional' material for making reeds. They give a warm, rich and woody sound and resonates perfectly. IF you can find a good one.
What every saxophonist struggles with is finding the consistency in a box of cane reeds. In a box of 10, maybe you will find 5 are great, 3 are average and 2 are just unplayable. That makes the unit cost of each great reed really high! The strength of each reed in the box can vary quite widely too.
However there are tools that can help you adjust your reeds, like a Reedgeek or a reed trimmer. However they are still time consuming and you can damage the reed if not used properly too.
Synthetic reeds are great because they are fairly consistent, don't react to the environment or moisture and last way longer than cane. Popular brands like Legere, Forestone, Fiberreed, Silverstein Alta and D'addario Venn have captured a big chunk of the reed market in recent years.
However, many people will feel that synthetic reeds tend to be brighter and less flexible in sound. They are also much pricier than cane reeds.
I see both cane and synthetic reeds as complimentary in my reed collection. I use them both, depending on the situation. To me, nothing beats a perfect cane reed. But when I can't find that, a synthetic reed can really save the day. Or if I'm doing lots of practicing at home, I might want to save my good cane reeds. When doubling on several saxophones on a gig or musical, having synthetic reeds on the sax you don't play much on might increase the ease and consistency of playing that sax.
This debate will never cease as long as saxophones are being played. So just find what works for you and go for the sound!