My wife has the Surface RT (original) and I have the Asus Vivotab RT.
We both updated ours to Windows RT 8.1.
There is good and bad to both. Where do I start.
Surface RT is really nice. It feels nice and solid but it's a little bit heavier, thicker and longer than the Vivotab RT. The Surface weighs 1.5lbs and the Vivo I believe was 1.2lbs.The screen on the Surface is I think like 1/2 inch longer (diagonally) than the Vivotab. They are both very responsive and easy to use. The Surface has a dedicated USB port but it has a funky power adapter. It took me 30 minutes to figure the "trick" to plugging it in. It's magnetic. But, it's nice as not to be able to damage the plug on the motherboard as you can with a traditional plug. The screens, other than the slight size difference, look identical and have the same resolution. Both bright and vivid. The specs inside as far as processor and RAM and HDD are also the same. There hasn't been any issues thus far reported by my wife after she updated to 8.1 BUT I use mine a lot more then she uses her's. She does use it, though.
The Vivotab RT is MUCH thinner than the Surface as discussed above, it's slightly smaller and weighs less. I purchased the keyboard with mine that turns it into a netbook. They keyboard has it's own battery and when both the keyboard and tablet are fully charge, you can walk out of the house with a netbook that lasts nearly 15 hours. The keyboards for the Surface are thinner and don't come with a battery. But, it makes it look like a tablet with a case which is nice. I had no issues whatsoever when I had 8.0 on my Vivotab. Going to 8.1 had a plethora of problems. It will freeze when going to sleep although there is an easy fix for that which has been very well documented. Some report the battery drains faster. Maybe like 5-10% faster. I have had A LOT of Wifi connecting/dropping problems. They seem isolated to places in public where you have to "accept" terms in a browser before you can proceed onto their network. The wifi keeps dropping and sometimes it refuses to connect. A few weeks ago I tried a multitude of things to fix this problem and to be completely honest, I am not sure exactly what I did to fix the issue. But, it WAS fixed. I then had a problem with keyboards not wanting to work with my tablet. I had made some registry hacks to disable some background stuff and I think those changes caused my new problem. I undid my hacks but the keyboard problem persisted. So, yesterday I performed a complete wipe and reinstall of Windows RT. AND behold I am back to my Wifi connection/dropping problem. Some of the fixes I tried last time didn't seem to fix it this time. Right now I have the device in Airplane mode (which still allows you to use the Wifi and bluetooth) and keeping it in Airplane mode has allowed the tablet to work on the Wifi at least for tonight. I am not sure if that will be a permanent fix. Oh, the Vivotab has a GPS and it's SUPPOSED to have an LED flash for the camera. Most people have not been able to get the flash to work so don't let that be a deciding factor. GPS works and is nice. (Surface does not have GPS). And the Vivo has a dongle/adapter for USB which converts the charging port to a USB port. The Vivo keyboard has a dedicated on board USB and its own charging port.
As I said, the Vivotab is very responsive and the screen is great and its VERY light and thin. If you're a techie guy that wants to work on 8.1 update issues, which can be resolved but not without some work, I would recommend the Vivotab. If you're not so savvy you might want to stick with the Surface RT.
Hope this helps.
**Update - First, I wanted to stress that the keyboard problems I was having on my Vivotab were undoubtedly caused by ME and in no way the fault of Asus or the tablet.
Second, as of today SO FAR I have not had any further Wifi connection/dropping/limited problems now that I have put the tablet perpetually in Airplane mode. Come to think of it, this MIGHT be what fixed it last time around before the reinstall of Windows RT. When I was having the connection problems before, I would attempt to disable and re-enable to Wifi on the tablet and just trying to do this would give the tablet fits (it would refuse to disable the Wifi). So, I would play with the Airplane mode switch thinking this might get the Wifi turned off. I would play with these switches quite often and I might have left Airplane mode ON and not even realized it. This switch is way above the Wifi information on the menu and unless you're specifically looking to mess with Airplane mode, it can VERY easily be overlooked. Since I don't have 3G on my tablet, Airplane mode can stay enabled all it wants for all I care. You can still enable Wifi and bluetooth after you enable Airplane mode. If I discover I am still having more wifi connection problems, I will update this again.