OK. I'll give my example. 2 LG G2s, same room, both Verizon. I will leave Advanced Calling/VoLTE out of this example even though it made a difference between my 2 phones, I also disabled that feature on both phones. Both phones had equal signal, but my calls dropped and data was slow. Other phone, no change, just worked, no problems.
You have tried reseating the SIM, no change. I don't buy into the idea that cases make that big of a difference in signal. XLTE is the only thing I can think of in my example, but it is only a theory and IDK if you have that in your area even though the S4 is capable. For whatever reason, my phone uses a weak XLTE band and the other uses a strong LTE band. Why my phone decides to run on the other band, IDK. I thought it might be that idea that new phones grabbed the weak 4G over the strong 3G like how it now grabs weak XLTE over a stronger LTE, or that maybe the towers decided to divide the phones based on location and traffic. Theories.
In the end though, my G2s don't allow network switching. Global, LTE/CDMA, and GSM.UMTS are all my choices. I cannot run 3G only, I cannot run the standard 4G bands, if XLTE is available the network pushes me over.
Perhaps you can change your radio Settings / More Networks / Mobile Networks / Network mode.
