I am watching Person of Interest season 4, episode 10 -- The Cold War, where Samaritan beat the Machine.
Why can Samaritan do things that the Machine can't?
Why must Finch, Root, Reese, Shaw and Bear hide?
The Machine is older than Samaritan, about ten years! So why isn't the Machine better than Samaritan?
Here is a (maybe non-complete) list of reasons why Samaritan seems to be better than the Machine:
This is due to the design decision Finch made. You can see that in the flashbacks of S04E05, where an old version of the Machine tried to kill Finch. There are two main design principles in Finch's mind when he created the Machine: The Machine must have a high moral compass (e.g. he teaches the Machine in S04E11 that no one - not even him - is more important than anyone else), and access to the Machine should be as restricted as possible, such that no one is able to use the Machine for his own needs, or even to manipulate the Machine. In the flashback of S02E14, you can see how angry Finch is at Ingram when he realizes that there is a backdoor to the Machine - which is the root cause for the numbers, the Ordos storyline, the virus storyline, etc.
In the season 3 finale, Greer convinces Control & Co that Samaritan has saved them all, while the Machine even did not see that event coming. Control has seen that Finch is willing to tell the public about the Machine, which makes Finch an enemy of the state: police are searching for them. The bigger problem is that Samaritan is searching for them, which is why they cannot attract attention (which is why the Machine can locate Shaw in S04E08). Finch & Co. are more or less the only antagonists of Samaritan, hence Greer wants them dead.
There is absolutely no single clue or whatsoever suggests that Samaritan is "smarter" than the Machine. Basically you are asking a question analogical to "why are the mafia leader smarter than the police chief?" That just simply isn't (necessarily) the case.
The machine loses to Samaritan constantly, in the real world and in Finch's Faraday-Cage'd simulation contests, due to one simply fact: Samaritan being evil and holding the entire human race in hostage while the Machine being good and trying to save all humans (though Samaritan doesn't understand the concept of evil nor does it perceives itself as evil). The playground is not levelled to begin with.
If we look at the case of S5E8 where Samaritan orchestrates a bio-terrorism attack using human body as the container of a highly virulent and highly contagious mutation of coronavirus (which Samaritan itself does not perceive as any sort of terrorism to begin with). Samaritan identifies a strain of H5N1 avian flu virus (supposedly from an ongoing outbreak in Asia) and a strain of human flu virus (supposedly from a pharmaceutical company's lab), and uses the human that carries the H5N1 virus (whose infection must also be Samaritan's work) as a petri dish, by forcing him making a huge detour and layover in New York and injecting him with the said human flu virus. All of these has a prerequisite — it needs a human agent immune to flu virus, i.e. Jeff Blackwell. Of course, that requirement is not out of the concern of the human agent's life, but to be able to kill 2 humans in a hospital locked-down due to a deadly flu outbreak.
We do not know how long does it take Samaritan to orchestrate all these, but we can infer from the crucial role Jeff plays in the whole attack, and that Jeff wasn't recruited until 8 months after his parole release. So, it must have taken Samaritan weeks, if not months, to come up with such a plan. But it only took the Machine minutes to realize Jeff's role, months ahead of Jeff himself realizing he's being turned into a murderer (and never realizing having become a hard-core terrorist), and it was work done during the initial chaotic period right after her first rebirth (S5E2) when she wasn't even considering context. As for the cure of the coronavirus, it took the Machine only a couple of minutes to design. If you look at the speed of designing a cure for COVID (both are caused by coronavirus) in the real world, the first two and only effective medicines Paxlovid and Molnupiravir took more than a year to develop and by far there is no cure for COVID. As cure is way more difficult to come by than to design a virus, we can be certain that actually the Machine is way smarter than Samaritan. But even the smartest wouldn't be able to save 7 billion hostages from a terrorist dedicated to wipe out the existing human race.
And go back to your question.
"Why can Samaritan do things that the Machine can't?" Oh, the Machine definitely can do those things but it wouldn't. Remember she put Reece on a flight to Rome? It could easily manipulate Reece to create another 911, but it wouldn't.
"The Machine is older than Samaritan, about ten years! So why isn't the Machine better than Samaritan?" The Machine is definitely better than Samaritan, way way way way better. Unless you think Bin Laden is the best person.
