Still, Rings does seem to be significantly different from Potter. The primary difference seems to be that Harry uses wizardry to pursue his own ends, whereas Gandalf and Frodo are self-sacrificing and use their supernatural gifts in order to do good and avoid evil.
That's the most I can say without having read the book and having read various synopses.
Much of the neopagan use of magic is the converse of this. It is frequently used to overwhelm, deceive and defile. In the Harry Potter series, for example, Harry resists and eventually overcomes Voldemort with the very powers the Dark Lord himself uses. Harry is the reverse image of Frodo. Rowling portrays his victory over evil as the fruit of esoteric knowledge and power. This is Gnosticism. Tolkien portrays Frodo's victory over evil as the fruit of humility, obedience and courage in a state of radical suffering. This is Christianity. Harry's world is about pride, Frodo's about sacrificial love. There is, of course, plenty of courage and love in the Harry Potter series, but it is this very mixing of truth and untruth which makes it so deceptive. Courage and love can be found in all peoples, even those involved in the worst forms of paganism. The presence of such virtues does not automatically justify an error-filled work of fiction. In Potter-world the characters are engaged in activities which in real life corrupt us, weaken the will, darken the mind, and pull the practitioner down into spiritual bondage. Rowling's characters go deeper and deeper into that world without displaying any negative side effects, only an increase in "character". This is a lie. Moreover, it is the Satanic lie which deceived us in Eden: You can have knowledge of good and evil, you can have Godly powers, and you will not die, you will not even be harmed by it-you will have enhanced life. There is so much that dazzles and delights in Rowling's sub-creation, the reader must exercise a certain effort to see these interior contradictions and mixed messages.
On the other hand, O'Brien says that the use of "magic" in Rings tends to diminish people/hobbits. I'm on page 50 now and I'm seeing that the ring diminishes whoever holds it. For example, Bilbo stopped aging and was tempted to hold on to the ring, even though he felt like he was withering away.