Problem 1: Physical DimensionsYou might notice that the HZ-519 is long and thin. Long and thin objects are delicate and require odd shipping boxes. As I, and at least one other Amazon reviewer found out, these things are easily bent in the middle during shipping as it is hard to protect, leaving me with some ugly and noisy bends in the metal. (The Amazon merchant did give me $24 back instead of taking the return). Also, while it has ample cord length, it is mounted at the right side, if your plugin happens to be to the left, the cord suddenly is no longer so long. Compare this with the squarish, solid HHP 1500 which fits in a box of normal dimensions, and whose cord can be used at a fair distance to the left and right.
Problem 2: Temperature ControlConsumer reports gives the HZ-519 high marks for its digital temperature control and timer, which are nice for a single use: set the temperature to maintain by pushing a few buttons, set the time to heat through a few more button presses. The HHP 1500 on the other hand has no way to set a temperature, you turn some knobs and it will try maintain a temperature: what temperature that is it doesn't say. You will have to futz with it over the course of a couple nights until you find a setting which is comfortable for you. But, and here is the big but, once you find a setting you like you are done. When you want to use the DeLonghi you come in flip the top knob on; with the Honeywell, you press the power button, press the "Temp/Timer" button to select temperature, hit a couple arrow keys; if you want automatically turn off the unit, that's more button presses. Every single session. There is no memory of the last setting. Yes, you do get a timer, which is nice, but you try setting the temperature on this thing in a darkened room, when all you want to do is crawl in bed. (There are versions of this heater with a display backlight, and with a remote control which would mitigate this annoyance.)
Problem 3: Being Nice to Your FusesThe HHP 1500 has two main power settings. One draws about 7Amps when powering its element, the other around 13 Amps, which spikes at over 15 when turning on causing my Kill-a-Watt to make a warning beep. The HZ-519 has one mode which draws around 11 Amps. A typical household circuit is rated at 15Amps. Now imagine having two bedrooms which share an electrical circuit, each with their own space heater. If you installed two HHP 1500s, that's fine, just keep both of them at their 7 Amp setting, and it will just take longer to heat the rooms. If you have two HZ-519s, pop goes the circuit breaker. This assumes the rooms in question are small enough to be heated adequately at the lower setting.
Problem 4: NoiseOK, I shouldn't criticize the HZ-519 because mine has a lot of twisted metal from the shipping incident, but it is quite noisy as it expands and contracts. The HHP 1500 is dead quiet.
Problem 5: Flexibility of PlacementThe HZ-519 is designed to be placed along a wall, while the HHP 1500 can be mounted on a wall or rolled into the middle of a room.
Energy UsageNot a big problem here, and impossible for me to compare. The HZ-519 in one bedroom has been using about 6kWH (about a dollar) a night keeping the room at a comfy temp over the chilly temp I keep the rest of the house, while the HHP 1500 has been draining about the same. I would be happy with nice thick blankets, but my wife wants the children warm when they kick off their covers.
SafetyThe HZ-519 does have such additional safety features as automatic switch-off when knocked over. You will have to judge how important this is for you.
In ConclusionIf you are in the market for an electric convection space heater, get the DeLonghi. Buy them in the summer when they are cheap, because they have really gone up in price since the cold weather set in. I paid $80 for a DeLonghi in late November, it's now mid-January and the same heater is $160.
[Updated with additional commentary after first posting]