For your question: Yes, it is a big upgrade from S4 to BA200. I own both two, BA200 now is one of my main portable IEMs with Fiio X3 + E17 or IPC + E17 and S4 was mainly used directly with my cell phone but only used for less than 10 hrs years ago. For TDK headphones, I also have BA100 and ST750 and I think these three are all good value headphones, at least worth its msrp price or even better after allowing relatively long time burn in and proper amping IMO.
What you should notice about BA200:
1. giving at least 20 hrs for burn in, otherwise you will think your S4 has better performance.
2. BA200 is NOT a good bass performer at all, I highly doubt those bass-head won't be interested in its sound. so I am hoping you've noticed this before your purchase.
3. if amping is possible, use a amp with your music players, a huge improvement after amping even just a portable amp can make a big step up.
4. for $140 it's a steal, I paid over $200 and I now still believe mine was a steal. BA200 perform like a $300 IMEs if amping properly IMO.
5. read more reviews to understand more about what you will get and you should be confident about the money you paid.
6. find a pair of right size ear tips and you will be good to go.
7. enjoy.