Some ships in Star Trek are referred to with definite articles. "The Enterprise", "The Defiant". Other ships are referred to without them. "Voyager", "Enterprise". Is there a pattern to or explanation for this behavior?
As far as I can tell (and yes, this is subjective since I wouldn't claim to have all instances where somebody refers to a ship fresh in memory), ships are sometimes referred to as if they were individuals and sometimes not. The form where a ship is called "The [Shipname]" is the unpersonal version (just as you would refer to an inanimate object) while the form "[Shipname]" is the personalisation (you wouldn't say "The Janeway" but you would say "Janeway" when addressing or referring to her).
The rationale why we might have heard "Enterprise" and esp. "Voyager" more often in ENT and VOY than we might have heard it in TOS/TNG makes a lot of sense if you consider the dependency of the crew on the ship. In VOY, the ship is the crew's life: They are completely depending on that ship. So, it makes sense to grow an disproportionate attachment to the ship and personalise it. Similar arguments apply to ENT, where in earlier episodes the ship was basically the only vessel in the fleet and later on, they were similarly far away as the VOY crew (relative to their respective warp capabilities).
Of course Picard's crew (esp. the engineers) were attached to their ship but they could afford to blow it up, as they constantly do in the films. They weren't that far away from home that losing their Enterprise would compare to losing Voyager (think about the tension in Year of Hell where the psychological stress on the crew even became visible as physical damage on the ship).
The name of the ship might have been chosen by the writers intentionally to facilitate this grammatically: A voyager might be a person, while enterprise cannot describe an individual person at all.
In Star Trek:
As far as how ships are called by crew or passengers, it was left up to the writers to decide how a ship was assigned the gerund (the) in conversation. In the real military, most ships were simply called by their name without any prefix unless there was a formal reason to do so.
In the modern Navy
Q: "What ship are you from?" A. "I am from the Nimitz." though it is also correct to say "I am from Nimitz." This second answer presupposes the person knows about the ship in question. Most likely the answer from one naval person to another.
Q: "What kind of ship is the Nimitz?" A. "The Nimitz is a nuclear-powered aircraft carrier. Its designation is CVN 68." This is a person who does not know what the ship is or what it's called. The answer would be considered a more complete or formal answer.
In the ship's newspaper and website correspondence the ship is simply called Nimitz in the short form and the USS Nimitz in the long form. Only in addressing for mail is it designated USS Nimitz CVN-68.
As far as personnel onboard are concerned she is "the ship". Informally, "the boat" or "our boat" (but only her crew would call her that).
When speaking to non-military personnel, she is "our ship" or a single word title, i.e. Mount Hood becomes "the Hood" USS Nimitz, becomes Nimitz.
For personal use I prefer just the name. For impersonal use such as describing the ship as an object, I use 'the USS name' or 'the name'.
some examples: For a captain on the ship, talking to someone outside the ship, it would be 'Enterprise to away team'. But if the ship was going somewhere it would be "the Enterprise will be there in 15 minutes".
"The USS Voyager is an intrepid class starship."
"Captain's log stardate 53116.2: Voyager has landed on an M-class planet, whilst we effect repairs and refuel."