As you present the distinction, it sounds like an objective statement that p, is an assertion that p (in other jargon, an utterance of p with assertoric force) which would commit the person who made the statement (the utterer) to the truth of p.
A neutral statement that p would be a statement of p that does not assert p. How to classify it further will depend on your taxonomy of speech acts and whether you're considering the neutral statement to really convey the same proposition. For example, take the statement:
(1) Earth has one moon.
If I assert (1), I am committing myself to the truth of (1). That would be an objective statement of (1), in your terminology (though I find the "objective" terminology here somewhat confusing since it has many different uses in philosophy).
If I make a neutral statement of (1), then I'm either uttering (1) non-assertorically, or I could be indirectly asserting a related proposition, e.g., I could be giving a belief report:
(2) (Some people say that) Earth has one moon.
In asserting (2), with the parenthetical bit being something understood from context, I would not be committing myself to the truth of (1).
See the SEP articles on Assertion and Speech Acts.
As I mentioned above, "objective" is used in a whole manner of different ways in philosophy. Off the top of my head:
- Truth-Aptness/Factivity: an objective statement is one capable of being true or false. Contrasts with, arguably, statements concerning matters of taste ("Pizza tastes good.")
- Reality of Objects Involved: an objective statement is one that is committed to the existence of the objects it refers to/quantifies over. Contrasts with, arguably, fictive statements or statements involving pretense ("Sherlock Holmes lives at 221B Baker Street." or "Suppose there is a round square.")
Given its ubiquity, I would avoid use of the term "objective" without explicitly defining it, unless the intended meaning is incredibly clear from context. If I didn't have to use the term, I'd probably try to find a better one that wasn't so widely and disparately used. In general, though, good practice dictates you define any (most) distinctions you make.