It makes it difficult to control registration information, and it makes it difficult for parties on the Internet to contact appropriate people in your business. Forms, designated contacts, and so on should remain with you, not the provider. Changes to the service usually take a while, and are a headache. For example, if the provider owns the routing equipment, they generally keep the password. It may take a couple of weeks to resolve problems.
Email forwarding can create a security breach.
If the provider forwards email, then all incoming and outgoing email is spooled on their machines where it is open to anyone who gains read access to the spool directory. It makes it difficult for the business to switch providers. This is particularly important if the connection is through a regional provider, who may develop money problems and fold, or may begin providing intermittent or unreliable services. The last thing to remember, and the most important, is that the Internet is still mostly run by a group of dedicated people, many of whom work for free, and who are always willing to offer advice and help.