As you all know, the main difference between authentic vs. graded materials is that in the latter, the materials almost always revolve around a particular structure that is presented to the student. For example, if the tense being presented is, say, "The Past Tense," every single speaker in the dialogs or even the texts given to the students are in that tense.
It seems as if there were no other tense in the whole world. In reality, when talking about the past, for example, native speakers may use a wider variety of tenses, sometimes even the present tense:
"Last night something very funny happened to me. I was walking down the street and suddenly a man comes and looks at me in the face and says: boy, you ARE ugly."
This is not uncommon in real life, but when it comes to graded materials, you will never find these types of situations that resemble real-life conversations. In spite of this, graded materials are very useful if you want to raise the students' awareness on a certain structures or patterns that may be important for them to learn. Every single context they see will contain many instances of the same structure which will enable them to make inferences on how they are used.
Authentic materials, on the other hand, are real in the sense that they are not created for students as the target audience but for native speakers. The obvious advantage, of course, is that by using authentic materials you present students with actual everyday language, just as it appears in real life. The main disadvantage of these materials of course, is that sometimes they are not teacher-friendly, and you may need to spend several hours reading or watching videos until you finally find what you need in order to use in your class. In addition, on many occasions in a whole context or situation you may find just one instance of what you need to present your students with. This could be overcome if you provide students with several situations in which the patterns appears, but again, you need to have the time to research and gather the appropriate materials.
If time is no object to you, you may well spend some time doing this research and you will soon find out that there's myriads of information out there that is terrific for your classroom! However, if you have time constraints, as most teachers unfortunately do, a balanced approach maybe the solution for you. You can use your graded materials to present the topic and later on you may find samples of that structure in authentic materials. Mind you, this will not be difficult due to the fact that whenever native speakers talk about something or write about something, they make use of nearly all tenses and structures of the language. You can even tell your class to go over some authentic texts, videos etc. and find similar constructions. Sometimes a combination of both approaches yields the best of both worlds. It is up to you to decide what could be the best for your classroom.