What's different between Premiere's own h.264 and QT h.264 encoders?
The encoder is always the same, namely Adobe Media Encoder. What you're changing is only the format, which admittedly is a bit confusing since that dropdown mixes up a couple of concepts which don't really belong together. If you select H.264 from the format dropdown, that's the video codec. Using that format selection, you will find a ton of options below, one of which is the multiplexer tab. This is where you can select what container format you want your video and audio stream to be mixed in (usually that's MP4). If you select the QuickTime format, you are always exporting a MOV file. The codec of the QuickTime format is variable, so you get a dropdown for that instead. 

So with both options, you will get an H.264 video stream, but mixed in a different container format. You can find more info on individual streams and containers in my answer here.

It's a bit confusing that you are selecting different things (codecs or containers) in the format dropdown, but since there are also completely different options such as exporting as an image sequence or exporting audio only, it's hard to come up with an interface that's neither terribly bloated nor missing important options.

From the Adobe Support Pages: 


  Some filename extensions—such as MOV, AVI, and MXF —denote container file formats rather than denoting a specific audio, video, or image data format. Container files can contain data encoded using various compression and encoding schemes. Adobe Media Encoder can encode video and audio data for these container files, depending on which codecs (specifically, encoders) are installed. Many codecs must be installed into the operating system and work as a component inside the QuickTime or Video for Windows formats.