Hi Mic,
There really isn't much in it, depending on the hammock. I find getting in and out of the zipperless net very easy, reach for the shock cord bottom open it up and swing your legs out. It allows a much easier adjustment of an under quilt when it's on as the quilt is inside of the net with you. I also leave the net on my hammocks and pack it all into one stuff sack, but it has the benefit of being able to be easily removed when you don't need it. Though as you know some hammock nets are able to be zippered off.
Installing an under quilt on an integrated net will be easier though. An integrated net will also have tie outs to spread the hammock out, so this may also make it easier for some to get in the hammock as the fabric is already spread out. Personally I dislike tie outs so it's a non issue for me but I can see how others would prefer tie outs.
Technically an integrated net may also offer greater resistance to the creepy crawlies, though I have yet to have anything unwanted get passed my net.
The main reason I use and make net-less hammocks with a separate net is for most hammocks that have an asym design, the net actually dictates the shape of the hammock, and that can mean it feels more restrictive for some people including myself. It also only allows you to lay on one particular angle. With net-less hammocks you have the use of the full width of the fabric and the shape of the hammock is dependent upon the angle in which you lay. This is the beauty of the Warbonnet Blackbird design, though I have yet to see one in person - I will this weekend though, the added fabric above the regular hammock fabric means that the lay is exactly like a net-less hammock. The net does not change the shape of the fabric upon which you lay like it does in a Hennessy or Dream or Hammeck hammock. Essentially if you remove the top fabric from a Warbonnet Blackbird you will be left with their Traveller net-less hammock, which is just rectangle shape. It's a stroke of genius really. In fact 99% of the hammocks are just simple rectangles which have a net added that changes the shape to an asym design, plus some well positioned tie outs to accentuate it.
Sorry that was a bit longer than I planned

. Essentially though I would say there is not a whole lot of difference but comes down to personal preference. I am going to start toying around with a integrated net hammock, and will put up a how to guide when I get it done. I would like it to be symmetric though for the reasons stated above.