To me the biggest advantage to not exclusively listening to internet based scanners is having the ability to listen to whatever systems/agencies/channels you want. With internet providers, you are at the mercy of whatever they want to monitor. For example, lets say you live in NYC in the confines of the 90 Pct. and you want to listen to what is happening in your hood, well guess what, there is no provider right now on Broadcastify currently providing that feed, so you would be regulated to listening to the citywide SO channel to maybe catch something going on on your block of a serious nature, as routine stuff would not be put out on the SO channel.
About the only thing I use internet provided feeds is to listen to stuff in areas I am way out of range of and that tend to have a lot of action, i.e LAPD, NYPD, Chicago PD etc. Internet feeds are also good for listening in on far away major incidents as well, such as tornado responses, civil unrest...etc.
For local everday stuff you can't beat having your own rig to monitor what you see fit as you see fit.