Someone once asked me how I got over the love of my life. My answer was simply, I am not over him, I just moved on. She was confused by my answer, as she considered both of them having basically the same meaning.
In my opinion, getting over someone is when the person no longer means anything to you. It is when they rarely cross your mind, and when they do, those thoughts and memories do not have any affect on you.
When you get over someone, you do not have anymore interest in their well-being or life. You basically do not care about what they are up to in life and whether they are happy or not.
When you have no love or hate feelings toward an ex, and their existence is irrelevant to your life, that is when you know you are over them.
When you move on, in my point of you, it means you are going forward in life getting used to the absence of that person in your days.
You still care about the person, you still think about them, however, you know that they do not belong in your life for one reason or another. You may still ask mutual friends about how your ex is doing in life, and what they have been up to, just because you still care and what to know they are doing well and are satisfied with their studies, work, etc.
When you move on, you might still have a weak heart towards that person, but that does not necessarily mean that you still want them back into your life as a lover.
Furthermore, you cherish the moments and memories made, and you are grateful for the experience and love that was shared.
Thinking about the person might still bring you some pain, but most of the time it will put a smile on your face. You relive the good times more than the bad, and that is why they will always be a small part in your beating heart.