Bureaucracy and red tape mean about the same thing.
Bureaucracy means as system where decisions are made through different departments. It generally means a form of government that has a lot of rules, regulations and paperwork. And as a result means slow and complicated procedures to get anything done.
For example, lets say that a fire in your home destroys your driver's license, birth certificate and social security card. With our system of bureacracy, getting these replaced is going to take a great deal of time and effort. First you have to contact your state's office of vital statistics and send a request (and probably a fee) for a replacement birth certificate. You have to wait until you get that through the mail. Then you have to go to a federal building that has a Social Security Administration office and take the birth certificate and some other proof of identification the government deems appropriate. You fill out a form for a replacement card and then you wait in line so a government clerk can enter the information into a computer and the Social Security Administration sends you a replacement card.
Then you have to go to your city or county's department of motor vehicles and fill out a form and stand in line and present your SS card and your birth certificate to get another license.
This whole process can take a couple of weeks and several hours just standing in line and filling out forms.
Of course it would be a lot easier to take whatever you need to establish your identity to one office, fill out one form, and stand in one line. That way you could get all the replacement documents the day after you lose them.
But because the division of labor isn't set up that way, it's a lot harder, longer, and complicated to get the job done.
Red tape is another an informal way of saying "bureaucracy".
The word comes from the French word for office (bureau--which also means desk) and -ocracy which means "power of". (Democracy means "the power of the people". "Theocracy" means the power of God, etc.) So you can think of bureacracy as the power of the office.