We need to distinguish between those terms as Soviet leaders defined them (i.e., in terms of Marxist-Leninist thought) as compared to the way that others might define them. The Soviets defined themselves as having achieved socialism en route to communism (which they never reached).

Marx and Engels believed that social, political, and economic arrangements passed through stages, having reached (in Europe) the capitalist stage. The hope of Marxist revolutionaries was to eventually reach the communist stage.

The (higher) communist stage is utopian in nature:

In a higher phase of communist society, after the subordination of the individual to the division of labor, and therewith also the antithesis between mental and physical labor, has vanished... and all the springs of co-operative wealth flow more abundantly – only then can the narrow horizon of bourgeois right be crossed in its entirety and society inscribe on its banners: From each according to his ability, to each according to his needs!

That is, at this stage, there is no more need for private property, society is truly classless, there is no need for money, and so on.

One cannot, however, jump directly there from capitalism. In the lower or socialist phase, society is in transition between capitalism and communism. The transitional phase is imperfect and temporary. Marx writes:

What we have to deal with here is a communist society, not as it has developed on its own foundations, but, on the contrary, just as it emerges from capitalist society; which is thus in every respect, economically, morally, and intellectually, still stamped with the birthmarks of the old society from whose womb it emerges... defects are inevitable in the first phase of communist society as it is when it has just emerged after prolonged birth pangs from capitalist society

There are some differences between orthodox Marxism and the Marxist-Leninist doctrines (as developed by Vladimir Lenin and the Bolsheviks) that guided the USSR. Marx believed that within the transitional phase, "the dictatorship of the proletariat" would be necessary to help society make the jump. In (higher) communism, there are no classes and all are equal (thus no bourgeoisie and no proletariat) but en route, it is necessary for the proletariat to control and remake society.

Leninism places more emphasis on the transition and its conflictual nature. Rather than acting a whole, the proletariat must be led a revolutionary party serving as the "vanguard of the proletariat" to move this transition to communist along. Under communist itself, all are equal and there is no need for any sort of coercion, but you have to use coercion to get there in Leninist thought.

The Communist Party of the Soviet Union and Soviet rule were instruments for the transition (that would eventually become unnecessary). We can break down into further sub-phases, but here is how the Soviet Constitution described the progress in 1977:

The Great October Socialist Revolution, made by the workers and peasants of Russia under the leadership of the Communist Party headed by Lenin, overthrew capitalist and landowner rule, broke the fetters of oppression**, established the dictatorship of the proletariat**, and created the Soviet state, a new type of state, the basic instrument for defending the gains of the revolution and for building socialism and communism...

The unification of the Soviet Republics in the Union of Soviet Socialist Republics multiplied the forces and opportunities of the peoples of the country in the building of socialism. Social ownership of the means of production and genuine democracy for the working masses were established. For the first time in the history of mankind a socialist society was created...

The aims of the dictatorship of the proletariat having been fulfilled, the Soviet state has become a state of the whole people... Developed socialist society is a natural, logical stage on the road to communism...

The supreme goal of the Soviet state is the building of a classless communist society in which there will be public, communist self-government. The main aims of the people's socialist state are: to lay the material and technical foundation of communism...

So, the Soviets had progressed through the dictatorship of the proletariat and into socialism, but this was just a "stage on the road to communism" which was the true goal (no private property, no class, no money, etc.). By the Soviets own account, they never reached that communist endpoint. Doctrinally, reaching "communism" would also have meant that the need for single party rule was over.

All of that being said, the terms "socialist" and "communist" have been appropriated by many others for often rather inconsistent purposes. But, under Marxist-Leninist thought, socialism was an intermediate step towards communism and the USSR never made it further than socialism. Today, socialism is often used to refer to an ideology that supports an expansive welfare state -- this is far removed from the Marxist-Leninist sense.

Sorry for the late reply. I’ve been writing this all day when I can find time at work.

This is a sensitive topic that gets to the roots of the biggest ideological schism in Marxist thought. Marx’s ideas are open to interpretation and have been interpreted many different ways. I’ll strive to present as close to an unbiased interpretation as I’m able but please feel free to point out anything you think is unnecessary commentary here. I’ll be providing sources for the Soviet union’s policy and disposition but not for what I perceive as the basics of Marxist theory.

In short, the Soviet Union was ideologically communist, or more specifically Marxist-Leninist and was ruled by the Communist Party. The government of the Soviet Union practiced what they called socialism, which they believed to be a transitional form on the road to Communism. Other communists have challenged the validity of their gameplan for establishing communism and some even went so far as to challenge the idea that they practiced socialism at all, most famously Leon Trotsky, who believed them to be what he called a “degenerated workers’ state.”

Communism as Marx and Engels defined it is a classless society where a state is no longer necessary and collectivized production is able to provide such a surplus of all life’s necessities that money becomes obsolete as a medium of exchange and everyone gets everything they need. They believed the establishment of such a society would be a natural consequence of the same historical processes that developed out of feudalism and capitalism.

In their view, the globalization of the bourgeoisie created a single common interest for the proletariat, who, comprising the vast majority of industrial society, had only to realize their power and assume control worldwide. The workers would then establish “lower state communism” (or “socialism” as Lenin and other later thinkers have called it) where state power and money or a money like medium of exchange would still exist and a “dictatorship of the proletariat” (most commonly read as not a literal dictatorship but a contrast to the “dictatorship of the bourgeoisie” that was republican governance) would restructure society to eliminate all class divisions and make all people equal. At the end of this project the state would simply “wither away” as humanity would have no more need for it.

The political realities in early 20th century Europe problematized these predictions, however. Revolutions across the industrialized world fizzled out or were successfully repressed. In Russia, which had not fully industrialized under Tsarist rule, would-be revolutionaries struggled to reconcile Marx’s predictions of revolution among the industrialized urban proletariat with their own desire to establish communism at home in a society that was still largely made up of peasants or peasant-adjacent workers who farmed the land with little mechanization or modern technology.

Several schools of thought emerged among Russian socialists during the waves of unrest from 1905 onwards. The orthodox school of thought was that Russia lagged behind western Europe in their historical dialectical progression and would thus need to have a bourgeois liberal revolution to overthrow the Tsar, establish a republic, and industrialize the country. Only once they were “caught up” to this part of the process would the workers truly have the potential to seize power and implement socialism.

Trotsky on the other hand believed that the historical process was better viewed globally rather than nation-by nation. The time had come for revolution worldwide and Russia could play a key part in it alongside the industrial countries. As long as the rest of the world stayed on their path to revolution, Russia could achieve and surpass the benchmark of bourgeois liberal revolution and jump into socialism all in one step.

Towards the end of the First World War, Russia’s government was on the verge of collapse and the Bolshevik vanguard was able to overthrow the Tsar once and for all. But the global revolution never came. Socialist and communist movements that seemed poised to enact their own revolutions fell apart or were crushed or fizzled out abroad. The communists were in uncharted territory. Beset by a world full of enemies and without a single ally abroad, party leadership practiced a kind of hybrid between these theories in which the workers’ state established industrialization programs designed to catch them up to where they needed to be to establish socialism, while at the same time turning inwards and focusing on protecting their dominion over Russia rather than expending resources to try to bolster other revolutionary movements.

These policies were later codified by Stalin under the theory of Socialism in One Country. The Soviet Union would later return their focus to fostering revolution abroad in one nation at a time, mostly focusing on underdeveloped and formerly colonized nations, but the dream of a single global proletarian revolution in all or most of the developed capitalist countries was dead and the dream of communism was put on hold. Khrushchev made it a goal in 1961 to establish communism “in the main” by 1980 with the slogan “communism in 20 years,” which famously prompted the retort “this slogan will survive centuries”

Sources:

Trotsky: 1905, Permanent Revolution.

Stalin : Foundations of Leninism

22nd Congress of the Communist Party of the Soviet Union: Third Program of the Communist Party of the Soviet Union