Why we need mail client ?
A Mail Client stores all the existing web mails in your PC or any other server depending on your choice 'imap' or 'pop' and you can access that any time but for synchronization to mails server you must have an active Internet Connection.
Some times we don't have Internet Connection in that case if we want to access our old mails then we can easily access them.
It may be possible that you are feeling very slow to Internet Connection then You can access mails on the basis of last synchronization.
I used two or three Mail Clients but now I am using 'Mozilla Thunderbird'. because it configures automatically , no additional setting is needed.
You can download it from it's official website.