The first kind of link you posted, with "/home/" in it, is the private URL of a file in a user's account. It doesn't offer access to the file unless you're logged in to that account.
The second kind of link you posted, with "/s/", is a shared link for a file, which does allow access to the file.
There isn't a way to get a shared link from the private URL unless you have access to the account.
If you do have access to the account, you can get a shared link as shown here:
https://www.dropbox.com/help/167
Or, using the API, using the /sharing/create_shared_link_with_settings endpoint:
https://www.dropbox.com/developers/documentation/http/documentation#sharing-create_shared_link_with_...
