If I understand your question, you're asking if sending a fax with your
computer attached to a phone line (which makes the computer the same thing
as a Fax machine) produces the same phone line bill as a stand alone Fax
machine connected to the same phone line.  The answer would be Yes, you're
paying for the phone line usage in either case.  You cannot use the Internet
to send Faxes unless you hire the services of an Internet Faxing company:

http://www.slipstick.com/addins/services/fax.htm#internetfax
Personal Fax Services - Internet Fax
http://www.savetz.com/fax/
FAQ: How can I send a fax from the Internet?

These may be more expensive to use than just paying the Phone company for
the phone line use.  The normal use for these services is for people that
have broadband Internet service but do NOT have a telephone line (folks with
Cable Internet service who only own a Cell phone).

Hal