The official rationale, according to Trump's presidential proclamation, is that the Centers for Disease Control and Prevention (CDC) could become overwhelmed "if sustained human-to-human transmission of the virus occurred in the United States on a large scale". It continues:
The World Health Organization has determined that multiple countries
within the Schengen Area are experiencing sustained person-to-person
transmission of SARS-CoV-2.
The Schengen Area currently has the largest number of confirmed
COVID-19 cases outside of the People’s Republic of China. As of March
11, 2020, the number of cases in the 26 Schengen Area countries is
17,442, with 711 deaths, and shows high continuous growth in infection
rates. In total, as of March 9, 2020, the Schengen Area has exported
201 COVID-19 cases to 53 countries. Moreover, the free flow of people
between the Schengen Area countries makes the task of managing the
spread of the virus difficult.
The United States Government is unable to effectively evaluate and
monitor all of the travelers continuing to arrive from the Schengen
Area. The potential for undetected transmission of the virus by
infected individuals seeking to enter the United States from the
Schengen Area threatens the security of our transportation system and
infrastructure and the national security. Given the importance of
protecting persons within the United States from the threat of this
harmful communicable disease, I have determined that it is in the
interests of the United States to take action to restrict and suspend
the entry into the United States, as immigrants or nonimmigrants, of
all aliens who were physically present within the Schengen Area during
the 14-day period preceding their entry or attempted entry into the
The reasoning exposed in the proclamation, then, is that due to the lack of travel restrictions between Schengen countries, it is far harder to manage the spread of the disease by controlling inter-Schengen migration. This factor is attributed by the proclamation as the main cause of community spread throughout the 26 member countries, and even goes as far as to tacitly blame the international spread of the disease on the area.
At a practical level, consider someone arriving to the USA from Slovenia, a member of the Schengen area. While Slovenia itself only has 31 cases currently, it is impossible for USA immigration officials to tell whether that person has just driven from Italy, which has over 10,000 cases. As a result, the Trump administration seems to have decided that the best way to deal with this issue is a blanket ban (with some exceptions, e.g. US citizens, travellers invited by the US to aid efforts against Covid-19 etc. See proclamation for full details) on travel from the Schengen area countries and visitors who have visited the area within the last 14 days, the incubation period of the disease.
Travellers from a non-Schengen area, however, seem to be presumed to be able to prove that they haven't been in any Schengen countries in the last 14 days by their passport records, and as such the level of risk they pose should be able to be more easily evaluated.
However, as pointed out in the comments, inter-European travel does not involve passport stamping in some cases, for example from Italy to the UK, so it will be difficult to identify these travellers practically. It is unclear how immigration officials will mitigate this.
The reasoning of the proclamation, however, remains that travel restrictions could be easily imposed between, to use our example, Italy and the UK, meaning that travellers from the UK are lower risk.
Update 14/03: The travel ban has now been extended to the UK and Ireland, which calls into question the original official rationale, but does not change it.