What is the difference between Mildred's and Montag's thoughts in Fahrenheit 451?
The point-of-view of the novel is 3rd person limited. Everything is seen through the perspective of the protagonist, Montag. Therefore, neither Mildred's thoughts nor the thoughts of any other characters are revealed. This was done in order to show how Montag alone perceives his surroundings. Most of his discontent throughout the novel comes as a result of the thoughts he has and his desire to know what others are thinking. He wants to know he is not alone. Instead of a "coming-of-age" novel, Fahrrenheit 451 could be classified as a "coming-of-knowledge" novel.
That said, consider this: Mildred is a shallow character. Purposefully so. When she is not sitting in front of her "family" (the soon to be 4-walled television programs) she is plugged in to her "mosquito" ear buds. It is likely that Mildred does not have any original thoughts, as she fills her ears and head with senseless noise all day (and all night) long. It is also no wonder when Montag tries to engage her in conversation that has an ounce of depth to it, she cannot respond. Consider when Montag asks her, "When did we meet, and where," she cannot even remember.
Montag's thoughts, on the other hand, progress throughout the book. The opening line, "It was a pleasure to burn," is then paralleled by his initial nervousness around Clarisse. He seems to take his job seriously and Clarrise presents a threat.
His initial sense of threat however progresses to his replaying conversations with her in quiet moments and eventually missing her. His thoughts then turn to the books he is hiding and the slight sense of fear, but the greater sense of purpose and power that comes with this secret. Once he begins meeting regularly with Faber we see Montag's thoughts take on a more philosophical air as begins to see himself partly as Faber.
By the end of the novel, Montag's thoughts have lost the fear and confusion they had at the beginning of the story. He makes decisions with confidence because he has gained a sense of knowledge throughout the book that affords him power, and his thoughts reflect this.