I've not got all that much experience with HDR, so hopefully someone else will jump in and give a better answer, but...
Neither Aperture or Lightroom have any native HDR capability, but you can use on of many 3rd party tools (Photomatix for example), and there are plugins for both tools that make this integration cleaner - but obviously at a cost.
If it's the HDR style you like rather than the actual need for the full HDR tonal range, you can achieve a very similar and clean result in Lightroom 4 alone by adjusting the 'Clarity' of the image (its just one slider to change along with brightness etc).
As for the choice - both Lightroom and Aperture are more than capable packages, and I believe you can get trial versions of both, so grab them and play with them and see which interface you prefer.
I said earlier that I preferred Lightroom, but I've fired up Aperture in an attempt to check a couple of things I was going to mention; I have to say that Aperture has come one quite some way since I stopped using it - it's got some really nice editing facilities that while they intimately can be achieved in both LR4 and Aperture, they way of getting you there is a little more direct in Aperture.
For example; if selective colour is your thing, you can achieve it in both packages - LR4 you mask an area and adjust the saturation; in Aperture, you can do the same, or you can go to a Black and White tool, and select a brush to paint in the effect directly - same result, but a little easier to find of you're just starting out.
For really complex stuff, you will probably want to switch to a dedicated editor such as Photoshop / Elements etc, but for the basic stuff (and basic actually is not all that basic anymore!), you can do it in both.