In my experience, tomatoes last longer in the fridge, in the sense of not getting molds and rotten, but the quality decreases a bit. Tomatoes seem to like the last bit of ripening outside of the fridge. Than you get the most flavor and the best texture.
When I eat tomatoes directly from the fridge, the tomatoes are too hard and tend to be a bit more sour. If they are softer because they are already older, the texture is more wattery and often there isn't much flavor left.
So if you have too much tomatoes to consume in a week, I would advice storing some tomatoes in the fridge for later. The ones that you are going to eat, I would always put them in room temperature a few days before.