I don't think either is necessarily better. I actually use both setups. If I don't feel like going to the gym, I use my home equipment. I can do most things with the equipment I have, but if I want to do a heavier workout, I go to the
local gym. At the local gym, there is a wider variety of equipment that would be expensive for me to add to my home setup.
If you are new to working out, I think home equipment is more likely to get used, and it can save quite a bit of travel time. The extra effort required to go to the gym can cause people to skip workouts. The exception would be a situation where you are meeting with a
workout partner (not your wife). If you have an obligation to meet them at set times each week, then you may be more likely to stick to that routine.