An anarchist by definition stands against all authority without exception, while a socialist by definition is simply someone who feels the means of production should be collectively owned. So, socialism is narrowly focused on economic issues, while anarchy is explicitly concerned with any and all social issues.

When a socialist also identifies as a libertarian, they're indicating that they're critical of the traditional authoritarian socialist states that have been so prominent in the world (the USSR, China, North Korea, Vietnam, Cuba, Yugoslavia, Zimbabwe, etc.)

But while libertarian socialists might reject one-party states, that doesn't mean they reject states entirely. A lot of them will support democratic states or other democratic forms of government. Anarchists, on the other hand, reject all forms of government.

Generally someone who chooses to identify as a libertarian socialist rather than an anarchist is making a deliberate choice to use non-committal language that implies they're willing to accept certain forms of authority. If they opposed all authority as anarchists do, they'd likely call themselves an anarchist.

There are various forms of libertarian socialism that promote a supposedly "libertarian" state, while there are other libertarian socialists who reject the state form, but embrace other forms of authority.

Communalists are a famous example of libertarian socialists who embrace various forms of authority including majoritarianism but stop short of supporting a full-blown state. But the form of government they do support greatly resembles states on a smaller, more localized scale. Communalists wholly advocate for government, majoritarianism, hierarchy and are probably best described as direct-democrats or socialist minarchists. Anyone claiming communalists are anarchists doesn't understand communalism or anarchy.

While a few anarchists might also choose to identify as libertarian socialists in polite company, the majority of libertarian socialists aren't anarchists, so anarchists would be better off avoiding the "libertarian socialist" moniker since all it really says about a person's politics is they like socialist economics but have an aversion to vanguard parties. Anarchy is a whole lot more than economics.

To identify as an anarchist is to take a strong stance against all authority, while libertarian socialism, democratic socialism and other such milquetoast labels take no such stance, leaving the door open to all kinds of authority, with the only real concern being democracy in the workplace.