Ocean currents bring lots of warm water from the south towards Europe making it quite a bit warmer than other places at similar latitudes. Cities in Europe are quite a bit further north than most people think.
Lots of other side effects as well would happen but be less known or surely predicted by models.
During the night, the lack of cloud cover also means there is less insulation to trap the heat accumulated during the day.
This can lead to hot days and relatively cooler nights, which can be characteristic of certain climates, like in arid or desert regions.
AMOC is part of a tremendous thermal-energy circulation system from the tropics to the arctic within the Atlantic Ocean, and tying to global ocean circulation patterns. Specific impacts are difficult to identify, but could well be hotter tropics (and likely: stronger and more frequent hurricanes and extreme storms), colder North Atlantic (including a loss of the moderating influence of the Gulf Stream on Northern European climates, hence potentially far colder and more severe winters). There are also likely impacts on marine life, agriculture, general wildlife patterns (plant and animal), amongst others. Regional sea-level rise is another possible impact (through thermal expansion if I understand correctly).
Effectively: a predictable and beneficial pattern to which human and nonhuman life and existence has evolved over many thousands of years, if not longer, may be on the verge of complete disruption. That's an event without precedent for on the order of 10,000 years (since the end of the last major glaciation period).
<https://en.wikipedia.org/wiki/El_Ni%C3%B1o>
More on prospective impacts of an AMOC collapse:
<https://en.wikipedia.org/wiki/Atlantic_meridional_overturnin...>
<https://www.nature.com/articles/s41586-019-1181-8>
<http://phys.org/news/2016-01-greenland-ice-sheet-affect-glob...>