Hey guys, thought I would offer some insight into this unfortunate issue.
A little background: I come from a family of Galaxy devices including the S3, Note 2, S4, and now the Note 3. It is a great device but it sure has its quirks. One of them being this WiFi issue.
At home, I have an ISP provided router/modem that all my devices connect to just fine.
At work however, we have a different story. The ISP at work handles things in a more secure fashion and you need to have a password in order to connect to the network. I am pretty sure that the connection type is WPA 2 (not that it should matter). I tried connecting to my work WiFi but failed doing so. First, I would select the WiFi from the list of available WiFi networks and then I would put in the password. After clicking the Connect button I am seemingly connected to the network, but after a while I get a message in the notification panel saying that I must login to the network. This was strange as I had already entered a password and I thought that was sufficient to use the network. So upon clicking the notification, it simply takes me to Google's mobile site and that is it! This was pretty weird as I was expecting to be taken to an online log in service for my work WiFi. I thought everything would be ready to go from there so I started to stream some music but was soon interrupted by yet another notification in the status bar. This time it said that the network authentication failed, and then another notification saying that the network timed out. I tried a lot of things including restarting the phone, and reconnecting to the network again, but it did not work.
This was reminiscent of trying to connect to the WiFi at my school. There, you did indeed need to use a web based login site in order to use the network. On my Galaxy devices, this was always a pain to do because the notification that usually tells you to sign into the network took a while to pop up. (I later found a fix in just opening up any webpage and then it will redirect to the web login page). After this, if I successfully logged in, the internet would again time out after a certain time and then I would have to repeat the process again. So all in all, it was a pain to use the WiFi at school (and now at work) so I just stopped trying.
Interestingly enough, this problem never existed on my Nexus devices. I found this out after taking my Nexus 4 to school instead of my Galaxy, and lo and behold everything was working properly. The notification alerting me to log into the network popped up almost immediately after turning on WiFi on my phone, and the connection was not intermittent at all - it was very stable and I could easily stream high quality YouTube videos without a hitch.
This got me thinking that perhaps this was just an issue with Galaxies, so instead of bringing my Note 3 into work, I recently started using my Nexus 4 again. As expected, the WiFi at work functions perfectly on my Nexus 4, as well as my Nexus tablet. Thus, this leads me to the conclusion that this is indeed a software issue. I have read through this thread and saw some potential fixes for the issue, but I cannot test them out at this time (since I am currently using my Nexus 4). I will stop by work tomorrow with my Galaxy Note 3 in hand and will test out the various fixes (dialing #0101 or something, as well as unchecking the network switch option). I will update when I get the chance.
As a sidenote, I was also encountering the same issue on the LG G2 (both at school and work). However, there were too many issues with that phone so I ended up returning it anyway.
Edit: So I tried out the disable WiFi Direct and Sharing solution, and it did indeed fix my issue. WiFi at work stays connected now without a problem. Although, it is a noticeably slower connection than on my Nexus devices. (I am almost positive this is just a Samsung issue at this point. Deal breaker.)