A Distinct Semantic Meaning
The ellipsis (…) is a punctuation mark. It is as distinct as any other punctuation mark and, as does any other punctuation mark, has its own meaning and uses. An ellipsis is not three periods. It looks like three periods and is more often than not, written as three periods—but it is not three periods.
Depending on the context it can have different meanings, but usually represents an omission. Other meanings may include an implied pause or silence.
The confusion over it's similarity to three periods isn't helped by the fact that in informal writing it is almost always written as three periods. In blogs, social media, text messages, emails, chat rooms and instant messages you will often see strings of periods ending a sentence.....
Usage & Guidelines
There seems to be a lot of debate on the correct usage of an ellipsis. Various style guides recommend different ways of forming an ellipsis. For example:
The Chicago Manual of Style and The Modern Language Association Style Manual recommend using three periods separated by non-breaking spaces, and surrounding spaces if not at the end of a sentence. The MLA Style Manual however differs from the Chicago Manual of Style in usage guidelines. If omitting material after the end of a sentence, MLA recommends including ellipsis (composed of periods and spaces) after the final period of the sentence (resulting in four periods), whereas the Chicago Manual of Style would only include the final period.
Various other style guides recommend other usages and there seems to be no consensus.
There are valid arguments for both using an ellipsis character and using periods separated by spaces. The article Dot Dot Dot . . . at creativepro.com puts forward an interesting argument:
Its problem is that it’s too small, constrained by the physics of font design to be only one em wide. […] If the idea of ellipsis points is to indicate an omission, they have to open a space wide enough to visually communicate a gap, […] a one em–wide opening in the text just doesn’t cut it
Visual VS Technical VS Semantic
The arguments seem to me to be visual vs technical vs semantic. Visually, periods separated by non-breaking spaces emphasise the omission or pause. Technically, periods separated by regular spaces may get broken across lines and an ellipsis character can get lost with translating character encodings. Semantically, the ellipsis has a distinct meaning from a group of spaces and periods.
In my opinion, using periods and spaces is nonsense. There is a distinct character for an ellipsis, so use it. A period means the end of a sentence, it is meaningless to end a sentence three or four times. Using periods (separated by spaces or not) just because it looks like an ellipsis is like using a 0 (numeral zero) as an O (capital O). But that's just my opinion.
In short—The character exists because it IS a distinct character and the benefits of using it are… debatable.