One of the main reasons why the USS Enterprise is so beloved, is that it is in itself a character in Star Trek. It's not just a vessel, but a character that viewers care about. I've been watching a lot of Voyager recently, and in the middle of an episode I came to realize that I don't feel that way about the USS Voyager. It never became a character to me. But why? What was different in the writing that made me feel for one ship, and not the other? If I think about the other Star Trek series, DS9 and Enterprise, I didn't connect with the Defiant either, but I did with the NX-01. Again, why?
What makes a starship not just a vessel, but also a character?
I think a lot of it has to do with the writing making the viewer feel that the ship is the characters' lifeline. The ship is there to keep the crew alive, and to protect them from harm. The 1701, and 1701D did that very well. The ships felt big, took a pounding, and rarely showed a lot of damage. The Voyager and Defiant however were constantly being beaten up, with bridge sparks and chunks flying. Bridge officers were getting hurt or killed on a regular basis (even it they were just no-name BOs), which made me think, "wow, for a state of the art starship, the Voyager sure does break a lot." Or "they should really think about rerouting all those conduits that keep falling on Sisco's head." When the ship is exploding all the time, you tend not to think of the ship as a character, but just as a device. It'd be like trying to get attached to Sulu if he was injured and in sick bay every episode.