Yes, HT will help with "background" progams just like multiple cores do, in the sense of being able to simultaneously run multiple apps.
However...
Technically speaking, Windows has been able to run "simultaneous" apps for a long time -- well before multi-core CPUs -- by essentially switching back & forth between the apps every so many clock cycles. We can't perceive it directly, of course; when a CPU can run billions of cycles a second, we literally aren't fast enough to see the OS switching from app to app every 10,000 cycles (or every 10 microseconds). However, what we do see is the slowdown in the apps themselves: if an app only gets 20% of the CPU's resources every second because it's sharing CPU resources with 4 other apps, it's going to take 5 times as long to complete its operations.
Multi-core CPUs take it a step further. Each core is separate from the other, & is treated as a separate CPU by the OS. So, if you have a quad-core CPU, & you have 4 apps running simultaneously, then they each have access to a full CPU's resources, & only have to share the other hardware/software in the machine (RAM, disk read/write access, Internet connection, etc.). So, for example, if you have 4 apps that all need 2GB or less of RAM, & you have a quad-core CPU with at least 8GB of RAM, then you can run all 4 apps at full speed; an older single-core CPU, even with the same RAM, would run noticeably slower, because the 4 apps would be sharing a single CPU, instead of each app having their own CPU.
HT bridges the gap between single- and multi-core CPUs. It's kind of a separate OS that layers on top of your normal OS, telling your system that there's twice as many cores as you physically have. Just like with physical cores, it lets you run additional apps on "separate" cores. But in order to do so, the HT "layer" splits the CPU's physical resources: clock cycles, L2/L3 cache, etc. So it doubles your cores from the OS's perspective, but each "core" now has the potential to run slower than normal because each only has access to half the normal resources.
Now, some apps (including games) have been coded to take advantage of multiple cores. Instead of the app running everything through a single core, it splits up its instructions & processes so that they can be fed through multiple cores simultaneously. It's not an automatic thing, though; it has to be specifically set up when the app is developed. HT is probably also similar, in the sense that your app would need to recognize that the CPU is equipped with HT.