Our trip leader in Puglia says the best way to get from Bari to Naples is by bus. From what I read, the train takes approximately 3 1/2 hrs and is more expensive than the bus which take approximately 3 hrs. Each of us will be traveling with a carry-on luggage and day pack. What are your thoughts? Thanks.

For that length of time, I might go for the train just because I can walk around on a train. But I'm sure either would be fine.

I would be influenced by whether the bus stops enroute for a break. Generally I prefer trains over buses if there is a choice.

If Naples city is your final destination, there is actually only one direct train: Bari dp 7.05am - Naples ar 11.15.
In all other cases you have to change at Caserta.
The bus ride is direct and takes 3 hrs (Flixbus) resp 3 1/2 - 4 hrs.(Marinobus)

https://www.rome2rio.com/map/Bari/Naples#r/Train/s/0

take the train and a picnic

Since the train takes a bit longer and is more expensive, I’d personally go with the bus, especially if it’s just a carry-on and a day pack. It sounds like the more efficient and budget-friendly option in this case.

The best way to travel around Italy is by train, it is much more convenient than by bus, especially since the bus can get stuck in a traffic jam. The price for a direct bus without transfers is 8.5 - 9.5 euros per person (time 3-4 hours, as luck would have it). The price for a direct train without transfers is 19.90 euros per person (4 hours). It is not much more expensive, but the ride is more comfortable and calmer.