It depends on the route and time of the year. Trainers are lighter but they are not waterproof, for some routes trainers are enough. Boots are traditionally heavier and are better suited for mountain terrain. There’s also walking shoes. It depends what you are used to and what you feel more comfortable walking in.
To make for a more pleasant Camino, it’s important to find shoes that fit well and are suited to the weather and terrain of the trip you’re planning. Here you have further information on how to choose the best footwear for your Camino trip.