There was an error while downloading the content. Please check the log for more details

This is the chat transcript between myself and an EA tech. I was having the same issue and this fixed it for me.

I am running Windows 8 but the process should be the same for Vista or 7.

info: Your approximate wait time is 6 minute.
info: Your approximate wait time is 3 minute.
info: Your approximate wait time is 3 minute.
info: Your approximate wait time is 3 minute.
info: Your approximate wait time is 3 minute.
info: Your approximate wait time is 3 minute.
info: Your approximate wait time is 3 minute.
info: You are now ready to chat with Ashutosh.
Ashutosh: Thanks for contacting EA Customer Experience, my name is Ashutosh. How may I assist you today?
you: Hi, I'm having trouble with the sims 3 launcher/store downloading purchases.
Ashutosh: Kala, I will try my best to sort it out for you
Ashutosh: May I know if you get any error message while running the launcher?
you: It does not give me an error message until I try to download the Roaring Heights world or the Boardwalk venue.
you: Then about halfway through the download the error message pops up saying:
you: There was an error while downloading content. Please see the Log for more details.
you: I have checked/Verified that the Sims 3 launcher is up-to-date. The version is current at 1.67.2.024001
Ashutosh: Which browser you are using to download the content?
you: Mozilla Firefox
you: I have no issues downloading smaller items like objects/outfits.
Ashutosh: I will provide you some troubleshooting steps, perform them.
Ashutosh: And your issue will be resolved
Ashutosh: 1.To fix this, you have to press the windows key and "r". type in regedit.

2.Then the registry window will come up. Expand HIKEY_LOCAL_MACHINE and then SOFTWARE and then Wow6432Node and then Sims.

3.If there are expansion packs in there that you do not currently have installed, then do a right click and delete.
Ashutosh: After this, restart your system
Ashutosh: Then download again
you: There are no expansions, only the base game. Which is all I have installed
you: delete the base game?
Ashutosh: No
Ashutosh: Just look if there is another expansion pack you do not have installed
Ashutosh: If you do not have any other expansion pack then we will proceed with another workaround
you: There aren't any listed
Ashutosh: Okay, let try clearing game cache
Ashutosh: Which operating system you are using?
you: Windows 8
Ashutosh: Go to C:\Users\"username"\Documents\Electronic Arts\The Sims 3
Ashutosh: username refer to your pc name.
Ashutosh: Delete the following files

Create-a-Sim Cache: CASPartCache.package
Build/Buy mode Cache: compositorCache.package
Mod or custom content cache: scriptCache.package
New Sims and skin cache: simCompositorCache.package
you: I have deleted them
Ashutosh: Delete all files within the folder "Thumbnails". Do not delete the "Thumbnails" folder.
you: Done.
Ashutosh: Now Exit the Origin if it's running
Ashutosh: And run the Firefox
you: Ok...
Ashutosh: Click Tools from the main menu at the top of the Firefox window.
Select Options (the Options window will appear).
Select the Advanced tab on the top right of the window.
Click the Network tab.
In the Offline Web Content and User Data section, click Clear Now.
Once the clearing is complete click OK at the bottom of the Options window.
Close all Firefox windows and restart.
Ashutosh: Then try to download the content again
you: Ok, now restart what? My computer, right? Or the Sims?
you: Ah, nevermind. restart Firefox
Ashutosh: Yes, and download the content again
you: So far, it is downloading without any problems; But it wasn't giving me the error message until it was about halfway done.
Ashutosh: You can stay connected if you want until it is halfway done.
you: It's about one-fourth of the way done.
Ashutosh: Okay, let's see if the issue still persist
you: That seems to have fixed the problem. It has made it past the point where the error message was popping up.
Ashutosh: Awesome
you: Thanks, Hang on till it's done downloading.
Ashutosh: You are welcome
Ashutosh: I appreciate your patience and cooperation
Ashutosh: Kala, Is there anything else I can assist you with while you wait?
you: I believe that was everything. IT just finished downloading!
Ashutosh: Enjoy gaming 🙂
Ashutosh: Thank you for contacting EA. Have a great day.Feel free to click on end chat.