I would theorize that it is because of the severity and the frequency of such tweets.
I don't think is being held to a higher standard, though it may just be a different group holding the president to the high standard this time. Obama obviously didn't get any passes here. Trump is just so in your face that I think we see it more often therefore maybe we perceive him as being held to a higher standard.
Or maybe, just maybe, it's because the Press is nothing more than an arm of the Democrat Party. Maybe they are down-playing the bad things one party does and exaggerating anything the other side does 24/7, whether or not it's really anything that rises to the level of "informing the public" vs. "guess what Trump just said this time?"
One relatively recent poll showed that the press gave Obama 42% POSITIVE news coverage, and Trump has received 5% POSITIVE coverage. Negative coverage was 20% for Obama vs. 62% for Trump - three times worse than Obama. I've seen earlier studies saying Obama's coverage was 80% positive compared to Trump's 80% negative -- polar opposites.
Even the New York Times published an apology after the election saying they were letting their biases drive their coverage of the candidates. Duh!!
That's based on the amount of time they spent on positive vs. negative stories, not the number of issues/topics. For example, if Obama was caught in a lie (like his Obamacare promises), that was played as few times as possible by the MSM. But if Trump was caught exaggerating the number of people attending his inauguration, that was the lead for almost every news channel on every program for 2 weeks -- all while ignoring anything good happening associated with Trump -- like record market levels and businesses bringing back thousands of good paying jobs.
Before you summarily dismiss my comments with "I don't believe the numbers", do some research.
https://www.washingtonexaminer.com/pew-trump-media-three-times-more-negative-than-for-obama-just-5-percent-positive