My answer - which I will explain further in my post - is that it is entirely likely that the topic of Leia is perhaps an even more painful subject matter for Kylo Ren to think about much less actually talk about... which is why he appears distant and far removed from her.
Though Kylo mentions little of anything of his mother in the movie, in the book, he reacts to her being mentioned by his own father... and with significant meaning.
“No, it’s not.” Halfway across the walkway now, Han continued to move forward, smiling. “Never too late for the truth. Leave here with me. Come home.” Without the slightest trace of malice or deception, he cast a dagger. “Your mother misses you.”
A strange sensation touched the younger man’s cheeks. Something long forgotten. Dampness. Tears.
“I’m being torn apart. I want—I want to be free of this pain.”
With the emphasis being made on the word "dagger" because it's explicitly being applied to, "Your mother misses you," and coupled with the description of, "long forgotten," the conclusion that Kylo Ren is distant from his mother can be reliably drawn...
However, at the same time:
Why describe reminding Kylo Ren about his mother as a "dagger"?
Why is Han's mention of said mother so specifically described as lacking both "malice" and "deception"?
My answer to the whys - which is my answer - is that far from being an unimportant figure to Kylo Ren, the topic of Leia/his mother is actually a particularly painful subject...
to the point that he's not only attempted to forget about her, but also if it wasn't Han/Leia's former? husband/Kylo Ren's father who was mentioning Leia to Kylo Ren, then the mention of Leia by anyone else would be considered to be related to an attempt at "malice" and "deception"...
or, in other words, a deliberate attempt at being cruel towards him (Kylo Ren) or an attempt to try and somehow trick him.
To further support this, regarding @DVK's quoted bit of conversation, Kylo Ren interrupts Snoke before anything more can be said regarding Leia.
And lastly, as to why Leia is likely a particularly painful subject... it's because Kylo Ren is - and had been - "torn apart"... and for what was/is likely a very long time.
Leia is 'Light'. Kylo Ren is 'Dark'.
Unlike Han who has become a smuggler again and who has, throughout canonical history, shown himself to have both 'Light' and more 'Dark' moments, Leia has canonically always been shown as a member of the 'Light'.
Add on the fact that Leia tells Han that she knew of Snoke's involvement (but didn't want to involve Han):
She sighed. “Many reasons. I was hoping that I was wrong, that it wasn’t true. I hoped I could sway him, turn him away from the dark side, without having to involve you.” A small smile appeared. “You had— you have— wonderful qualities, Han, but patience and understanding were never among them. I was afraid that your reactions would only drive him farther to the dark side. I thought I could shield him from Snoke’s influence and you from what was happening.”
... cut for brevity ...
“Always,” she told him. “From the shadows, in the beginning, even before I realized what was happening, he was manipulating everything, pulling our son toward the dark side. But nothing’s impossible, Han. Not even now, at this late time. I have this feeling that if anyone can save him— it’s you.”
And it becomes entirely likely that the 'Light' versus 'Dark' battle with Kylo Ren started since near the beginning which would have put him in a position of great conflict with Leia/his mother starting at the same time as she - because she never told Han - was the primary 'disciplinarian' figure.
Snoke tried pulling, Leia tried pulling back, fighting back, cajoling, whatever.
Whatever it was she did, Kylo Ren was being pulled one way by Snoke and being pulled in another way by Leia/his mother who was the primary disciplinarian figure and it was something that contributed to his "torn" state that he tells to Han.
Though Darth Vader is indeed Kylo Ren's grandfather by virtue of Leia, Darth Vader - not Anakin Skywalker who became Vader and who was later redeemed - and Leia were also complete opposites in terms of Force-leanings with Leia canonically never budging an inch from her 'Light' side leanings and with Darth Vader being a case of Light-gone-Dark and last minute redemption.