The simple reason why you should use a WORKING contact form instead of email address is that you get a lot less spam. You want your website to become more popular, but more traffic brings more potential spam. Here's how easy it is to get your email addresses from your website to email (and I'm not using some automated system):
The problem with contact forms is that when websites get older or some kind of change is made that's out of your control, those forms might have errors, and then you get nothing at all. It's worse to get nothing than tons of spam, but the best option is to just test your forms once a month and use those instead of publishing emails.
By the way, the website used in this example is not outdated or something I would recommend needs a full rebuild. Your site is maybe in the same boat: you have a decent website but you just get too much spam. Now that your emails are on spam lists, you will continue to get some, even after using a contact form, but at least you will stop most new spam from coming in.