Diskspace is the amount of space for files, databases and other content that your web hosting account or server provides. This is very similar with the disk space found on your computer, iPhone or any other digital device with storage capabilities.
Bandwidth is the ability to transfer files through the internet. The higher your bandwidth the more you can transfer. To put it in everyday terms think about the bandwidth of your high speed internet connection. The higher the bandwidth the more data you can transfer at one time. The more bandwidth your hosting account provides the more data you can transfer.
As an example, if someone were to download or view a 1 MB file from your website, it would use 1 MB of bandwidth.
Some hosts provide you unlimited disk space and bandwidth, so you do not have to worry about extra charges for exceeding a limit, while others allocate a specific limit.