What exactly would occur in the early stages of an impeachment inquiry that's not happening now?

[The Velveteen Ocelot]

8. What powers are those? They already have subpoena power.

The rules under which congressional committees operate have changed quite a bit since the Watergate hearings and subsequent impeachment hearings; the committees now have broader subpoena power than they did then. At that time they had to vote to authorize subpoenas; subsequent rule changes give them the authority to issue subpoenas without a formal resolution. It's not likely that a formal impeachment process would alter consideration of the most difficult issue, that of executive privilege. Here's a good article explaining how and to what extent (if any) the investigation process would be different if a formal impeachment proceeding were to be initiated: https://www.lawfareblog.com/what-powers-does-formal-impeachment-inquiry-give-house

One possible difference - at least in theory - is the power to obtain grand jury information:

...there is some historical precedent for the House judiciary committee to obtain such information from the court—most notably in the context of the Watergate impeachment proceedings. The relevant court opinion relied largely on a theory of inherent judicial authority, rather than an exception in statute, to turn the Watergate “road map” over to the House judiciary committee.

But on April 5, the U.S. Court of Appeals for the D.C. Circuit ruled that judges don’t have inherent authority to release grand jury materials and must instead rely solely on exceptions outlined in Rule 6(e). So if the committee wishes to access that information, Nadler will likely need to convince the judge overseeing the Mueller grand jury that release of materials to the committee is “preliminarily to or in connection with a judicial proceeding.” Bottom line: It is easier to argue that an open impeachment proceeding is akin to a “judicial proceeding” than it is to argue that any run-of-the-mill oversight activities are preliminary to a judicial proceeding.

It is also possible that courts would expedite their proceedings if a formal impeachment was in progress:

We think it is entirely possible—probable even—that judges would recognize the primacy of impeachment proceedings against the president of the United States and expedite consideration of such cases. The case of U.S. v. Nixon—in which the Supreme Court ruled that the president had to turn over the infamous Oval Office recordings to the special prosecutor—was decided just over three months after the relevant grand jury subpoena had been issued. That was a criminal investigation, so the analogy is not entirely apt, but we think it reasonable to assume courts would take a similarly expeditious view in the context of a subpoena issued pursuant to impeachment proceedings. Of course, it is worth remembering that the Supreme Court has never decided a case concerning a congressional subpoena for information issued to an executive branch official where the president has asserted executive privilege. In theory, the Supreme Court could decide the issue is a political question and leave it to the other two branches to sort out in some other way.

But the gist of this article is that the House wouldn't have significantly more power under a formal impeachment resolution than it does now.