I personally have a Vulcan .25 (Gen2, with lots of updates) and a Edgun R5 .25, not the R5M. In a nutshell I agree with Centercut, the wink goes to the Vulcan overall. Now with that said, it depends on the features that are most important to you as the shooter. If you want more detail read on.
First up the good things about the Edgun R5(M) series overall.
Easy to work on, Simple to breakdown for cleaning barrel and other areas and parts. Good job ED! R5 .25 has an incredible moderator system, very quiet and very, very powerful capability, built very well. Trigger is a true hunting trigger and if you want a competition style trigger, you will be disappointed. With that said I do have my Edgun trigger adjusted down to 1 pound 5.5 ounces. The gun is not designed to have the trigger this light. It seems happiest at about 2-3 lbs of trigger pull. Mine is adjusted so light that if you take the safety off, it will DISCHARGE, every time. So I don't use the safety feature on the Edgun, I simply de-cock. Did I mention the power available in the Edgun R5 .25? This gun is actually a .30cal with .25 barrel and breach, so shooting the 34grain pellets, is no problem, STOCK, no mods required. The gun is well balanced and easy to shoulder and it feels light in your hands. Weighing right at 10 lbs with rings and scope 8-32x50 mounted.
Now the bad? about the Edgun R5.
Only comes in wood stock (Ed has video showing a "plastic stock" coming soon). The wood is slippery to the touch. There is a lot of post trigger pull "jump" to the gun. The gun is pellet fussy, now with that said, I am still experiment with this facet of the gun. My preliminary results are pointing the gun is more Pellet head size fussy than mfg fussy, I am still working on this as part of the speed that the LW (ed design twist rate) barrel likes the best. Accuracy is very dependent upon a clean barrel, the right sized head of pellet, and the speed. To adjust the Hammer spring, you have to take the stock off. The bolt action, can become rough, requires frequent lubing, roller ball guide on bolt is simply held in place with a spring and therefore can not "roll" as the bolt is cycled, lube helps. I have shot hole in hole at 50yards with 33gr. To simply turn to a secondary target with the same pellets same air fill, same everything but shoot at 100yards to only see my pellets spiral at some distance between 50yds and 100 yards. 5inch groups at 100yds with the spiral. (working on fixing this, I feel confident it has the potential for better accuracy, but with a hunting trigger, and post trigger action (jump) of the gun, accuracy will be impacted.) The trigger adjustability is minimum, you can adjust the stop after the second stage, in other words how much trigger travel after the sear releases, and the "Offset" of the sear engagement, which relates to the pull of the second stage break point. Travel of the first stage is long. This is a hunting trigger, so says Ed, and he has no interest in changing that part of the gun, as I was told by Ed directly. So, with that said, it is a good hunting trigger.
The good about the Vulcan, let me try to match the same parameters I discussed about the Edgun above.
The Vulcan comes in a nice tactical black synthetic or a wood with either a curved but or a "block" butt on the older Vulcans. The Vulcan is not as pellet fussy. It does require cleaning but not as dependent and fussy as the Edgun about a dirty barrel. The Gun is Accurate!! My trigger is currently adjusted to have a simple glass break 2nd stage of 9.7 ounces, with very little first stage. To adjust the Hammer spring, simple turn the hex nut on the back of the gun. You don't have to remove the stock. The post trigger pull action of the gun is very smooth, no jump like the Edgun. Then again the Vulcan is designed to shoot 25.3gr and the Edgun R5 is designed to shoot the 33.9gr, so they are putting out different amounts of air, so the post trigger action will vary. The Vulcan like the edgun R5M has a center mounted action and is ambidextrous. Personally I prefer the bolt of the Vulcan or if you prefer you can have the lever action on the Vulcan. My Vulcan weighs in at 10lbs 8 ounces with a 6-24x50 scope and bipod. Accuracy strongly goes to the VULCAN.
I did not mention my experience with Post buying customer service about the Edgun on purpose. So let me say this about the Vulcan customer service, it is EXCELLENT!!! I have contacted Evgeny Pigrov of Airgun Technology on several occasions, probably 3 years after my purchase of the gun, not expecting much support since the gun was out of warranty. He not only responds, but answered all my detailed questions and helped me troubleshoot, but sent me a new regulator (improved version of the older design), degassing tool, new manometer, new o-rings, new springs, I probably left something out of the list, but at NO CHARGE. Now, that is customer service. So the wink goes to the VULCAN in customer service, not to mention you can also turn to your retailer for help as well, but when the owner of the company takes care of his customers like Evgeny half a world away, that means something.
The Bad about the Vulcan
A bit more of a pain to work on, small fasteners, but realize I am picking on the Vulcan here on purpose. I have replaced the regulator and found it pretty easy, but the wink goes to the Edgun on ease of breakdown for repair or mods and cleaning. The Vulcan needs a moderator. It has 5 moderator cones included in the barrel shroud but I put a Donny FL SUMO on mine and it is very quiet, but the wink actually goes the the Edgun R5 .25 for being the quietest believe it or not, even with the Sumo on the Vulcan. It's close though.
So, IN CONCLUSION, the wink ;-) goes to the VULCAN family overall, but the Edgun is a nice gun and personally I own both and will continue to own both.
These are only my opinions and experiences, someone else might have different views and opinions and I truly respect their experience and opinion might be different. I would enjoy hearing them. I just hope my 2 cents might help someone else down the road. Hope this helps. Happy shooting!