Content warning: Politics. Questionable takes? (click or activate to expand post)
I have seen the mistake of confusing the two concepts at least one time, and it is probably not uncommon in cultures that are rather far away from both of them (such as, for instance, at time of writing, the U.S.A.). This is not meant to be an opinionated post advocating for either, and it is also not extremely detailed or accurate (if you want that, read the Wikipedia pages on "Social market economy" and "Socialist economics"). It instead aims to point out the main differences in a rather compact manner, so when I see this kind of confusion the next time, I can quickly copy-paste the link to this rather than having to explain it all by myself.
Social market economy
Social market economy is a typically capitalist style of economy where the state, instead of choosing one of the extremes of either free market economy or planned economy, tries to combine advantages of both. Companies are still allowed to compete, they and their means of production are owned by private actors, and the state does not interfere with any specific companies in general where not needed due to emergencies or illegal behaviour. However, the so called "social system", consisting of insurances for healthcare, pensions, etc., as well as other social services like money for those without work (in Germany this is known as "Hartz IV") is mandatory and financed by public money. Examples of social market economies include most Western European countries and Canada.
Socialism
Socialist econonomy is mainly based on the ideas of planned economies where companies and means of productions are owned by either the government or some kind of other collective, such as a worker union. In contrast to communism, where rewards for work are based on the needs of people, in socialism the rewards are based on the abilities of people. Historically, socialism has often been advertised as a sort of transitional step from capitalism to communism. There is normally some kind of social system as well, however, dealing with unemployed is typically handled by the government creating a place for them to work at, rather than simply paying them based on the condition that they do not have any occupation that pays them enough wage for them to survive on. Goods are typically manufactured for use rather than profit, and priced accordingly. Most of the examples of socialism are historic and they rather failed. These include the Soviet Union, which later dissolved and expanded into the USSR, Cuba, and some East Asian countries like China or (North) Korea.