Since you are wondering which are the hardware differences between the DISH Network Sling Adaptor and the Slingbox PRO-HD, I will recommend to check the following article or contact Dish for more product information on the Sling Adaptor itself.
For the Slingbox PRO-HD, you will need a router and an AV source to stream the content received by you TV provider. If the network meets the requirements, you should not have any streaming issues on a remote device.
Feel free to use this link for more product information:
I hope that helps.
What is your upload speed for you home ISP rated at? It's NOT the download speed, but rather the upload speed at the slingbox host location that is the main factor as to most stuttering / buffering / loading issues. You could verify this by checking the stream of the slingbox at your host location using a wifi or lan connected computer/phone or device. If t plays fine then it's your upload speed to remote locations that is the issue as I mentioned above).