Barack Obama accomplished what Richard Nixon wanted to do.

Richard Nixon was forced from office because he tried, and failed, to use the IRS against his political enemies and because his "Plumbers" tried to wiretap the DNC.  For this he was forced to resign his office.  Both issues were part of the indictment drawn up by Congress to impeach Nixon who resigned before being impeached.

Barack Obama was much more successful.  Whereas the IRS refused Nixon's orders to target his political enemies, Obama's IRS cheerfully took up Obama's cause and managed to effectively neuter the Tea Party movement by denying them tax exempt status.   the Tea Party was an effective grass roots movemetn that was stalled by lack of funding as a direct result of the IRS actions against them.  The IRS Director of the IRS Exempt Organizations division Lois Lerner,  refused to testify before Congress and invoked her 5th Amendment right to refuse to answer on the ground that her answers might incriminate her.  She retired and the Obama Justice Department took no further action.

The proximate cause of Nixon's demise was "Watergate," an attempt by the "Plumbers" to plant listening devices in the Watergate HQ of the DNC to find out who was leaking negative information to the press.

The Obama administration succeed where Nixon failed.  They were able to listen in on many conversation by and about members of the Trump team and leak stories to the press that implied Trump was conspiring with Russia to win the election.  The leaks were designed to imply the Trump was a Russian puppet but failed to stop his election.  There is still no evidence that there was collusion between Trump and Vladimir Putin although Democrats have continued to make the charge.

There is no question that Trump and his staff have had their telephone conversations intercepted because numerous stories have appeared in the press either quoting, paraphrasing or characterizing those conversations.  Unless the American an British press are lying about the leaks they have received from members of the Obama Administration, there is no question that Trump's calls have been intercepted. The only question of fact is whether these phone calls were tapped as an "incidental" byproduct of foreign surveillance or if the members of the Trump team were the targets of these interceptions.  

But the fact is that the acts that forced Nixon to resign were accomplished by Obama with no effect on him finishing his term.

There are two primary reasons for the difference in the result.

1. The press.  The press hated Nixon and were ready and willing to do anything to remove their political enemy from office.   First. he was a Republican.  Second, he was an anti-Communist who defeated one of their favorites, Helen Douglas, for which they never forgave him.  Third, he never hid his disdain for them.  Contrast this with the press' adoration of obama, a man who they declare spotless in scandal department.  
2. Republican morality.  Republicans, by and large, observe an Judeo-Christian morality.  Nixon could not be removed from office unless Republicans voted for his impeachment and conviction.  When the "Watergate Tapes" were played, Republican leaders decided that the ethics revealed by Nixon reflected a morality that they could not support in a President.  They formed a committee to tell Nixon he had to resign for the good of the country.  Contrast that with the tribal ethics of Democrats during the impeachment of Bill Clinton.  Democrats in Congress exhibit a tribal loyalty to their party which differs from the religious based "universal" morality that Republicans - by and large- adhere to.    This is not to imply that Republicans are Boy Scouts, but they understand that there is a moral standard that is above loyalty to the tribe.