note: not asking specifically about Trump & Cuomo.
A recent US president was impeached twice in their first term by the US House and both times acquitted by the Senate. The second trial from February 9 to 13 2021 was close to the end of their term and since the election was lost they wouldn't be returning.
If I understand correctly, arguments against a conviction included "the sole remedy for impeachment and conviction is removal from office and Trump was no longer holding the office" while for a "conviction" included preventing them from running again.
In the impeachment of a New York State governor would a conviction also prevent them from running again?
This question is inspired by (but not limited to) something in the recent CNN news item New York lawmakers suspend Cuomo impeachment investigation:
New York state lawmakers will halt the impeachment investigation against New York Gov. Andrew Cuomo, who resigned this week and will leave office on August 25, state Assembly Speaker Carl Heastie announced on Friday afternoon.
In a statement, Heastie said Cuomo's resignation effectively eliminated the central question of the probe -- "whether Governor Cuomo should remain in office" -- and that Judiciary Committee Chairman Charles Lavine advised him, with the guidance of counsel, that the state constitution "does not authorize the legislature to impeach and remove an elected official who is no longer in office."
which brings up this governor's resignation as a reason to suspend the impeachment.
Question: In general, what is the differences between the impeachment and conviction of US president and New York State governor once they are out of office?
Do they carry the same or different repercussions in terms of punishment, deterrence (for the recipient and for future candidates) and future career, financial and political opportunities?