Star Trek / Captain Kirk / Conflict Between Feeling (Kirk) and Logic (Spock)

As you said, it is a common thread woven into Star Trek and so occurs in the Voyager series as well (Voyager is my favourite Star Trek series). Here's one of my favourite scenes regarding this topic:

Star Trek Voyager, Season 2, Episode 19 (“Lifesigns”, Stardate 49504.3).

Context: The Doctor is not a human; he is the Emergency Medical Hologram (which is active, because the regular medical officer has died). Kes is a humanoid alien, who is close friends with the Doctor. Danara is a physician, who is physically dying, but a (healthy) holographic copy of her consciousness has been created.

DOC: I'm running a level two self-diagnostic right now. I'll be there as soon as I'm finished.
KES: Why are you running a diagnostic?
DOC: I've been experiencing periodic lapses in concentration and difficulty handing objects. There may be a malfunction in my tactile acuity subroutine.
KES: How long has this been going on?
DOC: About two or three days.
KES: Since Danara came aboard.
DOC: What's your point?
KES: Maybe she's the reason you're feeling this way.
DOC: I fail to see a connection.
KES: Maybe you're attracted to her.
DOC: I told you, my programme's malfunctioning.
KES: Romance is not a malfunction.
DOC: Romance is not part of my programming.
KES: Your programming's adaptive, isn't it?
DOC: Yes.
KES: Then I'd say it's adapting.
DOC: What if I don't want it to adapt right now?
KES: Why wouldn't you?
DOC: Because I don't like what's happening to me. I'm used to being in control of my faculties, confident of my decisions. But lately, whenever Danara's programme is deactivated and I should be concentrating on my work, I find myself thinking of nothing but her.
KES: Why don't you reactivate her?
DOC: Because whenever I do that, I suddenly feel unsettled, unsure of myself, and I have no idea what to say. Why would people seek out situations which induce such unpleasant symptoms?
KES: Because when the other person feels the same way you do, it's the most wonderful thing in life.
DOC: Suppose, hypothetically, of course
KES: Of course.
DOC: I wanted to pursue that possibility. How would I proceed?
KES: You'd have to tell her how you feel.
DOC: That's it? Just tell her how I feel?
KES: Otherwise you may never know how she feels.

Star Trek Voyager, Season 6, Episode 18 (“Ashes to Ashes”, Stardate: 53679.4).

Context: Seven Of Nine, who was Borg drone since childhood, is now mostly human again (as much as is possible). She tries to supervise children, which were likewise Borg drones, but for a short while only. Icheb is one of the kids. Seven struggles with understanding them. She has already been taught that children need recreational activities, but the concept remains utterly alien to her. In the mess hall, she instructs the children:

SEVEN: I've allotted one hour for recreational activity. There's no time for irrelevant conversation. Adhering to our schedule will allow us to use our time more efficiently. Fun will now commence!

[A minor disagreement among the playing children ensues.]

SEVEN: You are encouraging disorder.
ICHEB: And you never let us do what we want.