Up front: Yes, a long password with symbols has more average entropy than a long password without.
Ultimately password security choices are left up to the people who design the systems that you use. There may be other additional security features that they use to secure your account information other than a username and password combination. IP/Network information can be used liberally to corroborate your identity with your logins. As an example, let's say you access your account from a public library on a different ISP than you normally use. Your bank may freeze your online banking account and contact you by phone at the phone number you have on file to verify your identity. It's just one example, but you get the idea.
Some may have other reasons for banning the use of symbols in their passwords due to poor data verification that could expose their systems to attack. Maybe it's just cheaper to do that instead of actually hiring a bunch of people to fix the problem for good. Who knows?
You do have a choice in the matter, and if that limitation bothers you, find a new provider for the services that you use on their site. You can always make that choice.