The 5 most common nouns in a sentence starting with The distinct... are advantage, feature, difference, features, nature.
The 5 most common nouns in a sentence starting with The distinctive... are feature, features, character, characteristics, characteristic.
It's unrealistic to suppose that the two usages are somehow referring to different "types" of features, so I suggest the two adjectival forms are to a considerable extent interchangeable.
But note that distinct tends to mean clearly discernible (sharply defined, not out of focus), whereas distinctive is more often used to mean striking, noteworthy and/or clearly distinguishable (easily identified as "different" to other things of the same general class).
So Hitler's distinctive toothbrush mustache, for example, is thus referenced because it stands out from other mustache styles, but a distinct echo is one that can be heard clearly (we're not normally comparing it to other echoes! :)
For OP's exact context, there's no doubt that the most common adjective is distinctive long neck (easily distinguishable from the necks of other animals), but there are a number of written references to a distinct long neck in Google Books, so I'd say that's a "valid, but less common stylistic choice".
There are also a number of written references to a distinctly long neck (unquestionably long), which in some contexts is a perfectly reasonable stylistic choice.