The downside is that you have a slow EC2 GPU with 4 GB RAM. Conv nets that take 3 weeks on a EC2, will take less than 2 weeks on a GTX980. If you run large conv nets, the 4 GB can be limiting (for example on ImageNet or similarly sized data sets).
Another point is that is more convenient to work on your own desktop and you can run multi-GPU nets, which is not possible on EC2 because the virtualization kills the memory bandwidth between GPUs.
If you think about it, over the long term a personal system will just be more cost efficient (you can keep a good system for years). So for deep learning researcher and those that apply deep learning this is just the most cost effective option.
A example calculation: You can buy a faster system than a EC2 for roughly $400 (GTX 580 + other parts from eBay). Together with electricity costs thats about 1 year worth of EC2, or 2 years worth if you use deep learning sporadically. A high end deep learning system will be about $1000-1400, which is about 3 years worth of EC2. So a EC2 makes good sense, if you use deep learning only sporadically and work with small data sets. If you use deep learning heavily, want a faster system or want to use multiple GPUs, a personal system will be better.