MightyCambo You can definitely download the world as many times as you want, on as many computers as you want. But there are a couple reasons this might not work on one system. First, please clear your browser cache, or use a different browser; Firefox tends to work well with the Store.
If that doesn't help, did you install teh game in the same way on both computers, or did you use different methods? For example, one might be a disc install that you later updated with the EA App, while the other was only downloaded straight from the EA App.
Separately, you can transfer the .Sims3Pack from your old laptop to the family computer, and it might work, depending on whether the install types match or are close enough. The file will be in Documents > Electronic Arts > The Sims 3 > Downloads on your laptop, which is also where it needs to be on the other computer in order for the launcher to read it.
You can use an external drive or USB stick to transfer the content, or you can use a cloud storage service like Google Drive or OneDrive. If you upload the file to cloud storage, zip it first: right-click the file and select Send to > Compressed (Zip) file. On the new computer, right-click and Extract.
