When a company files its articles of incorporation with the state, it will start doing business as (d/b/a) the company name ending in either "Corp." or "Inc." which serves to legally inform the public that the entity in question is not a natural person, and thus has limited liability. Technically within the United States, "corp." and "inc." designate the exact same thing. A corporation, in its simplest terms, is just a legal entity that stands apart from any natural person.
"Inc." and "Corp." convey the same basic fact about the company organization, it is not considered proper to use them interchangeably. Because a corporation has to designate the name that the entity will be using to conduct business, it is necessary to pick one or the other and stick with it. That means if you go with “Inc.”, then all your official letterhead, correspondence, domain names, business cards, and any other company related documents and sales collateral would have to include the use of “Inc.” rather than “Corp.”.