There are some goods and bads about the new vs older Mac Pros. My old one is a 2006, which is of course much different than a 2010, so take that in to consideration. I also have a 2013.
Having internal storage is great. Yes, I use thunderbolt drives with my 2013, and I can move the Logic library to the TB to help free up space. The TB is great for files, but there are a fair amount of things that need to stay on the internal or that don't work as well on an external, such as dropbox, AE cache, bootcamp, etc. And for me there is Carbonite where I have unlimited backup on all my internal drives. It is a culture shock switching from the luxury of 4 drives, which can be dedicated to certain causes to managing one small SSD. Also, I think latency is a concern with audio recording to thunderbolt or USB 3 drives.
Right now, any upgrades to the 2013 are enormously expensive. I don't see that changing because the components are specific to that line of Mac Pro. You can't use a standard SSD or RAM. You can't use a flashed PC graphics card because the physical structure of the card is different than for PCs. So the reasonable longevity of the newer Mac Pros is suspect.
All that being said, my 2013 blows my 2006 and my Macbook Pro out of the water with 3D processes and rendering. The GPU is underutilized right now, mostly because 3D apps are just starting to take advantage of it. That will change in time (but of course Apple will make sure the 2013 is obsolete by then). It is important to note that the GFX cards are not gaming cards and may be better suited to some applications over others. They are also not NVIDIA and cannot take advantage of CUDA processes, which are required for certain things like AE raytracing.
As an extra, the newer Mac Pros are much more transportable. If you anticipate needing it in multiple sites, it does make a huge difference. It also means they are easier to steal, and it requires additional hardware to lock them down. They are super quiet too, so that can help if recording.
I think if I had a 2010 and were not doing 3D, I'd keep that running with upgrades a while more. I think the newer Mac Pros will have less of a life span, so getting one now means having to get another one in about 3-4 years to avoid problems. I'd be inclined to see if there is a 2019 refresh. By that point, the whole Mac Pro line might change.