What is the difference between a pigeon and a dove? Biological it is not that different. In any encyclopedia you will get the word dove when looking up pigeon and the other way around.
However the way we look at the two birds is very different from each other.
The Dove is perceived as something that resembles Peace, pureness and love. It is also the messenger. It spread its white wings and carries important letters from one country to another. In a way it almost seams like a savior, and that is perhaps the reason why it is seen as an analogy for the Holy Spirit in Christianity.
The pigeon on the other hand is by the majority seen as the bird that is as close to a rodent as any bird can be. They live in the city and eat the food us humans leave behind. Often they are scruffy and lacking a crawl, and we consider them filthy and annoying.
In Danish language the two are not separated. They have the same name: due. The way we distinguish them from one another is by telling their place of habitat as a part of their name. For example skovdue (forestdove) and bydue (citydove). We associate due with two different things just as in English speaking counties, and yet we have only one word for both.
A reason for the two different names in the English language is that they have their roots in different language families. The Dove (taube, duif) originates from the Germanic languages and the pigeon (pijon, pipion) originates from the romance languages, whereas Danish only have roots in the Germanic language.
So what is better?
Usually a word is invented because there is a need for it, because we are trying to define a subject within the frame of language. It would seem stupid to have to describe the bird every time we had to mention it, instead of giving it a name. And maybe it is not a bad idea to use two different names for what certainly has two different definitions, a negative and positive.
Just like “a little angel” and “a brat” are two different ways of describing a child.