In one sentence: The Rust Programming Language (TRPL) is for beginners who know programming and Programming Rust (PR) is for intermediate programmers.
Now for a more in depth answer. TRPL is what made Rust 'click' for me coming from an application developer's perspective. Try to focus all your energy on completely understanding the memory model. Create a text note called 'rules.txt' and write down all the memory rules found in the book (ownership rules, borrowing rules, etc). Try to create a personal project to experience the memory model while reading your 'rules.txt' when you encounter compiler issues. I cannot say this enough, becoming a productive Rust programmer is dependent on understanding the memory model.
For PR, I felt as if the book's purpose was to give an in depth explanation to most of the features found in Rust. If TRPL is the foundation, PR is the building. I reread this book from time to time because it has so much knowledge on a variety of Rust topics and it serves as a sort of status marker of my Rust knowledge. The book keeps me humble and inspires me to learn as much of Rust as possible. It would be a good investment to buy this book if you are driven by fiery curiosity and want to understand as many Rust features as possible.
In conclusion, no matter what learning material you choose please remember the following sentence: it pays to understand the memory model of Rust. All the pain you hear about in learning Rust is from people who ignore the previous sentence. I should know since I used to be one of them. Once you understand how the memory game is played, you can finally experience what all the hype is all about. Good luck!