Many I meet are very salesy. Makes me wonder if they are truly after my best interests? Banks seem so aloof and laissez-faire. The kind of attitude that sends the message that they don't care if I take my business elsewhere (which leaves me with the impression that they are the best deal around).
With a bank, you get one and only one bank's rate. With a broker you get several bank's rates. Moreover, brokers I have known also can get deals with other sources of money besides banks such as private lending.
Basically it is a matter of how much you value your time. I think if you really wanted to you could apply to each and every source of funds a broker would and get roughly the same deal in the long run*.
However if you have something better to do with your time than to fill out the same kind of applications time and time again, study the laws and rules for the different products lenders offer, and then negotiate deals with the different lenders, then brokers can be a worthy expense. Brokers also have inside and firsthand knowledge with lenders, so sometimes they can swing favors to close a deal or get extra time.
When I did my shopping, I just put brokers and bankers against each other from all over town. I don't think it hurts to let them know you are shopping them around, but I wouldn't share the rates you get from one with the other.
*Brokers might get a better deal because they take a sales load from a lender, but brokers also add cost so I imagine it is a wash, but that is just opinion with no facts to back them up.
Just to add to the existing points with my own experiences in the UK:
Communication: some banks are better than others when it comes to the customer being able to contact the right person. With a broker, you have a single point of contact, and it's their job to talk to the bank.
In addition, if the broker already has an established relationship with a bank, there's a good chance they will be able to speak to relevant people there directly in a way that a customer cannot.
For example: one bank I dealt with required me to start proceedings by speaking to their mortgage manager, in person, at a local branch. When I subsequently had questions, they were never available on the phone. When I then employed the services a broker, the broker was happy to do everything by phone/e-mail/post, and was able to bypass the branch altogether and speak directly to the right people elsewhere.
Wiggle room: On a related note, a broker may have direct access to the mortgage underwriter, who is not someone that a customer ever gets to speak to. Assuming the broker understands your particular circumstances, they may be in a position to persuade the underwriter to use his discretion if you're seeking to borrow a little more than might otherwise be possible.
Experience: if you've never been through the process before, it can be useful to have someone to act as a guide and deal with some of the admin (and stress!). They'll also be in a position to suggest relevant insurance products (though hopefully without being too pushy...).
Complex arrangements: if your purchase is at all out-of-the-ordinary, and involves multiple banks or other organisations, having one person whose job is to manage it all is very useful.
Note: my experiences are with a brokerage company who charged a fixed fee for their services. I'm not in any way affiliated to any brokers; just a satisfied customer!