Hard links can work only on the same file system. They are basically different names for one and the same group of data (file). I.e. if you create hard link to a file, and then you delete the original file, the link will continue to exists, and the data will be accessible trough the hard link. The disk space is released when the last hardlink is deleted. Actually, there's no difference between the original file and the hardlink you created later. They are interchangeable. After creating a hardlink, there's no more "original" and "link".
Softlinks can be used between different file systems, i.e. a link can be created in filesystem2 to link to a file/dir on filesystem1. Also, if you remove the original, the link becomes broken, as it points to a undefined location.
Yes, you can think about softlinks as a "shorcuts" in windows, but they are not exactly the same. Shortcut in windows is a file by itself, and if a program can not interpret them, it can not use them.
Here is a nicely pictured article on the topic :)