I would like to give some brief background for my question to avoid answers that explain the difference between fully connected nets and ConvNets.
I completed the first 3 courses in the deep learning specialization from deeplearning.ai (Coursera). Simultaneously I've worked through the first two home work assignments from Stanford's CS321n course for the Spring of 2017, as well as Michael Nielson's book on deep learning.
I believe I have a reasonable understanding of the architecture of both types of networks, and how to write them in python/tensoflow. I also implemented my own very simple fully connected feedfoward net in C where I was able to get 98% on MNIST.
I've also done some reading about this question in terms of which type of data is better suited for a ConvNet, and the answer I've come up with is similar to:
If the order of the input features matters to their meaning, then a ConvNet will work best, but if you replace the position of one column with another and the meaning is still intact, then a ConvNet will not work.
This means that spreadsheet data (if I've understood correctly) is not suited for a ConvNet.
Unfortunately I don't know any deep learning practitioners to ask this question: are there fully connected deep neural networks in the wild that are currently in use? The best example I can think of is financial data. If that's true, what are some others?