Well this was a learning experience for me but I eventually figured it out. I'll explain my process here so that it's easier to know how to figure this stuff out on your own (BTRFS documentation, as I'm sure you found out, is relatively incomplete for the time being).
At first I thought that creating the subvolume was an ioctl with a handler that didn't do any capability check (which may or may not have been a security issue depending on whether there was some logic to it) whereas deleting it was modifying the metadata directly (and thus the user might require CAP_SYS_RAWIO to work properly).
To verify, I cracked open the btrfs-utils source code and this is what I found:
Create subvolume, cmds-receive.c Line 180:
ret = ioctl(r->dest_dir_fd, BTRFS_IOC_SUBVOL_CREATE, &args_v1);
Delete subvolume, cmds-subvolume.c Line 259:
res = ioctl(fd, BTRFS_IOC_SNAP_DESTROY, &args);
Well, that ain't helpful, they're both ioctl's (interesting side note: "snapshot" is often used interchangably in the source code with "subvolume" for some reason). So I went to the kernel source code and found both handlers in fs/btrfs/ioctl.c.
Eventually, I traced it back to btrfs_ioctl_snap_destroy() and on line 2116 :
if (!capable(CAP_SYS_ADMIN)){
Specifically, this is a check if they don't have the capability but if they do have it, the logic skips straight to performing the operation. The body of the if statement checks to see if it's regular user who's the owner of the subvolume's inode and the USER_SUBVOL_RM_ALLOWED BTRFS option is enable it continues executing the handler. If they don't have either the ioctl handler exits with an error.
So it looks like destroying a "snapshot" (aka "subvolume") generally requires a user that has CAP_SYS_ADMIN (or for USER_SUBVOL_RM_ALLOWED to be enabled and the user "owns" the given subvolume). Great, what about creating a snapshot/volume?
The handler for the ioctl appears to be btrfs_ioctl_snap_create() this handler appears to contain no call to capable() directly or indirectly. Since that's the main way access is brokered I'm taking this to mean that subvolume creation always succeeds. This explains at a functional level why you're seeing the what you're seeing.
I can't speak to why this is considered desirable outside of BTRFS's main use case being with a server with restricted user access. That's not sufficient but I'm not seeing any code to actually stop the operation. If you can't find an answer to why that is (and you care to have it) you may have to ask on the kernel mailing list.
Conclusion
My research seems to indicate that anyone can create subvolumes but in order to delete a subvolume you either need to have CAP_SYS_ADMIN or it needs true both that the calling user is the owner of the subvolume inode and USER_SUBVOL_RM_ALLOWED enabled.
The subvolume creation doesn't make sense so I'm probably missing some indirect way that the operation is denied since that seems like an easy way to DoS a system.
Note: I'm not in a place where I can verify this functionality but once I get home I can set if setcap magic works how this predicts.
I think this is where you went wrong:
If I try to run:
btrfs subvolume snapshot /mnt/disk/root/snapshots/2015-05-01 /mnt/disk/root
It creates the new subvolume as /mnt/disk/root/2015-05-01 instead of
replacing /mnt/disk/root/.
btrfs subvolume snapshot is used to create a snapshot of the first argument, and it place it in the directory given by the second argument. It sounds like you're expecting it to replace /mnt/disk/root instead.
Before you try to overwrite the /mnt/disk/root subvolume, you'll need to move or delete it (for example mv /mnt/disk/root /mnt/disk/root-backup-during-restore). Then do:
btrfs subvolume snapshot /mnt/disk/root/snapshots/2015-05-01 /mnt/disk/
And then:
mv /mnt/disk/2015-05-01 /mnt/disk/root
Best Answer
Turn off and delete the swapfile.
Linux 5.x introduced official swapfile support for Btrfs. Unfortunately, this support appears to be, ahem, incomplete. I suppose what is happening is that a subvolume which contains an open non-COW file cannot be snapshotted (snapshot? snapfleshwounded?). On some level this makes complete sense. On another, wtf. sigh.
If you want to use a swapfile, I think it should be fine to create a separate subvolume, however I have not tested this. Otherwise, you really do need to delete the swapfile before making the snapshot otherwise the file will get reflinked, stop being non-COW even though you marked it as such, and swapon won't work.