When you need to share files to a few people only, then use the Share function. With "Share", you can share to DriveHQ members or non-members; just select a folder in FileManager or in the Show Folders page on the website, click on Share. You can then enter your friends email addresses or DriveHQ username. You can set different permissions; moreover, a folder can be shared multiple times to different users with different permission. Just make sure you change the Share Name.
If you want to make your files available to anybody on the Internet, then use Publish. Note you must be a True or Premium Account to publish files. The process is very similar to Share.
Now the main differences are:
(1) If a folder is published, you can easily find the Publish URLs to the folder and files in it. You can use Publish URLs in other web pages to link to the folder/files. No logon is required for accessing published files.
(2) If you want to share a folder with "Write" permission, you must use Share and set the correct permission level. To upload/modify/delete files in a folder shared to you, you must also be a DriveHQ member, and the folder must be shared to you with Full Access right. You should logon your own account and access the folder shared to you in the virtual folder "DriveHQ Share".