I am running Maverick. Also, the SparkleUpdatedLog.log has the following...
2014-07-08 19:19:02 +0000: ===== HazelHelper.app =====
2014-07-08 19:19:06 +0000: Extracting /var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/Hazel-sparkle-3.2.7.dmg as a DMG
2014-07-08 19:19:08 +0000: copyItemAtPath:/Volumes/04E7232C-BA69-42C0-9D0D-2E5CEE2B817E/.DS_Store toPath:/var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/.DS_Store
2014-07-08 19:19:08 +0000: copyItemAtPath:/Volumes/04E7232C-BA69-42C0-9D0D-2E5CEE2B817E/.fseventsd toPath:/var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/.fseventsd
2014-07-08 19:19:08 +0000: copyItemAtPath:/Volumes/04E7232C-BA69-42C0-9D0D-2E5CEE2B817E/.TemporaryItems toPath:/var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/.TemporaryItems
2014-07-08 19:19:08 +0000: copyItemAtPath:/Volumes/04E7232C-BA69-42C0-9D0D-2E5CEE2B817E/background.png toPath:/var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/background.png
2014-07-08 19:19:08 +0000: copyItemAtPath:/Volumes/04E7232C-BA69-42C0-9D0D-2E5CEE2B817E/Hazel.prefPane toPath:/var/folders/p3/ry589b013nxc76lff_9qgpg80000gn/T/Hazel 3.2.7 1/Hazel.prefPane
2014-07-08 19:22:10 +0000: Sparkle Error: An error occurred while installing the update. Please try again later.
2014-07-08 19:22:10 +0000: Sparkle Error (continued): Couldn't move /Users/Brian/Library/PreferencePanes/Hazel.prefPane to /Users/Brian/.Trash/Hazel (1077).prefPane.
It appears to me that Hazel can't trash the older version. However, I have tried installing the update by deleting the panel and I can do that but I still get the same message when I try a fresh install of Hazel.