Pros - You have complete control over who can access the mail accounts. For example, I have my server locked down to a single IP address that can access the mail system. So, nobody but me is going to send mail from my server.
Cons - If you don't have the server setup properly you can run into mail being rejected by recipient systems. Be sure to have the server host name set properly, the reverse DNS MX record set properly and if possible activate spf, DKIM and DMARC on the server to lessen the likelihood of mail being rejected.
And if you're running it off of a residential connection, you will most likely have problems with people receiving mail simply because you can't set the reverse DNS (or even have it changed).
I don't use web based email clients. Never have, never will and don't allow them on the sites I host on my servers. So I can't speak to security of them. But think about it, if it's web based it can be hacked or compromised.