Specific moments occur in episode 42, where, after becoming an Idol Singer, overcoming hardships and traveling to America to find him, Mitsuki finds that her childhood friend Eichi died a year and a half ago. This is either better or worse in the manga, in which Eichi died in a plane crash and Mitsuki knew from the very beginning that he was dead, but was in serious denial and became Full Moon to find him anyway.
Then there's the last episode, where you can cry in despair for the first half and in happiness for the second.
Izumi's backstory in the manga where we find out that he killed himself when he was six standing in a railroad crossing, because he felt that his mother didn't want him. It turns out that his mother let him go because she felt like she wasn't able to love him.
The manga has the revelation of why Mitsuki can see the Shinigami. Eichi's ghost never left her, and it's through him that she can see them.
Mitsuki: Between [Meroko's] hand and mine... is Eichi.
The ending of the manga, with Mitsuki's song.
Takuto singing. It's so painful to know that he could do that forever and nobody would ever hear him.
When you think about, Fuzuki calling adult!Mizuki Hazuki is pretty sad. She pushed her daughter away by saying how she doesn't like her future son-in-law instead of accepting their relationship. So when she sees her granddaughter grown up, she's reminded of what she's lost.
Meroko's backstory is pretty bad, too. Dreaming of love and finding someone she likes, then finding out she's suddenly engaged-to-be-married to her best friend's fiancÚ (and hearing that said best friend was going to marry the guy she liked), getting almost raped by him and even seeing proof of her best friend and crush being together drove her over the edge...
Of course, this makes the reconciliation she has with Fuzuki in the manga all the more heartwarming and having you cry tears of joy.