Ok, I've done some searching and here's what I found...
It seems that a lot of people on Sprint experience random data connection drops. I'm not sure if Sprint is known for this (I'm on AT&T myself), but it could account for all the negative "reviews" I keep seeing about how crappy the Sprint data network is. Some people even claim the reason why they are the last carrier left to offer fully unlimited data plans is because the data network is not very reliable and that's one selling point to keep people around. Take that for what you will, but anyway...
When it comes to data drops like this, my first response (as an computer guy) would be to start with a fresh restart...in your case, yank the battery and go from there. Once restarted, the first thing I would do is to check for all updates. And I'm talking about more than just the software updates. Check for Profile updates and system updates as well as any others you find in the updates section of your phone.
SMS and Internet
I was searching to see if there were any issues with sending/receiving SMS (which are not sent using the data connection) while using the internet connection (Skype in your example) and one person wrote this over on the Sprint forums:
"The Galaxy S3 does support Simultaneous Voice and Data. However, it is not a guaranteed feature. It is reliant on network availability. If there are not enough network resources on the tower you are connected to the data connection will be turned off. Voice services always have priority if there is a conflict."
I know you're on the S4 now, but if this problem is not your phone, then there could be some truth to what he's saying. If this is the case, there could be some issues with your local cell tower. Are you experiencing this problem everywhere you go?
In order to test the network's data reliability, you probably want to do various tests in very different locations. If the problem occurs in places as far apart as say 10-20 miles, there may be an issue with the phone itself. I've seen reports online from people who have had defective phones and once they got exchanges everything was fine. However, if the problem only seems to happen within a specific area, you may want to inform Sprint as there may be some technical issues with one or more towers in that location. As another quick test, get with some other people that use Sprint and see if they're having the same issue.
I'm assuming your comment about wanting to go back to your old S3 meant that you didn't have this problem before, so that leads me to believe it's a phone issue...not that all S4's will give you this problem, but maybe just yours. My best advice is to walk into a Sprint store and try duplicating the problem. Even if you can't, you're within your trial period still and should be able to return the phone for another regardless of the reason.