Please document your personal experiences with the HR44/HR34 here. This should act as a point of reference that DTV customers can use to confront the DTV CSRs when trying to upgrade from the HR34 to the HR44.
My story:
I have just upgraded from the HR34-700 to an HR44-700 and it is tremendously faster - the menus pop up instantly now, where with the HR34 they were always delayed and when scrolling on the HR34, if a movie banner came up the menu would stutter and get slow (very annoying) on the HR44 it smoothly scrolls right through them. From an interactivity standpoint, and in the words of a DTV supervisor, the HR44 is "night and day" faster than the HR34.
Another major improvement on the HR44 over the HR34 is the speed of FF/REW/PLAY/PAUSE - in the HR34, it would always be behind the interaction - when I hit FF then PLAY, it would hesitate and sometimes would lose it's place in the content. These problems are completely absent on the HR44.
Bottom line: the HR44 is MUCH faster than the HR34 on everything but channel changes which are only slightly faster. As to whether it's worth the upgrade and what DTV say about it - that's a different story you can see profiled in the post on "DIRECTV whole home DVR service" -> "HR44 Genie is 300% faster than HR34, why does DTV deny it?" DTV personnel are trained to say that the HR34 is the same speed as the HR44, despite all of the evidence to the contrary. Also, if you are on the Equipment Protection Plan, DTV will not upgrade you from an HR34 to an HR44. I was successful when a retention supervisor gave me a $290 account credit so that I could buy the HR44 on Amazon, but it was a tough fight.