There are a lot of factors at play. First of all, the Soviet plan to eliminate religion was in effect only as long as it was contributing to the national project. This meant that state efforts to suppress religion were put on hold during the Great Patriotic War, as religious rhetoric/ imagery contributed to national pride.

In the interim, we have to look at how they handled their suppression--it was fairly haphazard, with the occasional church demolition (to be fair, including some high-profile demolitions like the Cathedral of Christ the Saviour being bulldozed to eventually build the Palace of the Soviets but in the end just becoming a community swimming pool) or arrest, public ridicule frequently but irregularly, the only things being fairly constant being book bans. But many churches still survive, both because the suppression was never systematic, and because one can only ramp up a campaign so much against an institution that was once important to the state. The Russian Orthodox church was more or less allowed to continue its existence. Many Soviet citizens also still openly celebrated holidays like Easter and Christmas, even if they somewhat subdued the religious aspect of it. In sum; the communists may have wanted to eliminate the church in the end, but "the end" never came close to happening. Their suppression was pretty hamhanded and never a top priority.

Islam was wholly different, owing to its prevalence in border republics. In the case of Islam, Soviet authorities might promote atheism in the region but were cautious to anger a population they saw as vulnerable to ideological conditioning against the state and didn't fully understand.

I cannot speak much to other religions except that they did not enjoy the relative comfort in their societies that Islam and Orthodoxy did. Judaism is an exception; anti-semitism existing in Russia still existed under the Soviets. They created a Jewish Autonomous Republic but look for it on the map and it's quickly apparent how serious they were about protecting Jews by giving them a homeland.

Now, the second part of your question. As Russian Orthodoxy existed as an institution throughout the USSR, it still had a clergy, facilities, etc. Now it is important to note how lost many Russians felt after the fall of the Soviet Union. In the aftermath, the early 90s were a chaotic time for Russia with improperly adapted economic therapy, political upheaval, loss of structure. A lot of people had voids in their lives (the Soviet regime tried create a place for itself in many aspects, down to entertainment and culture), and so in the absence of state suppression of religion, and the need for unity, clarity, and answers, church attendance shot back up. The new state also began to coopt the Russian Orthodox Church (not on the level that the Tsars did, but drawing from the same blueprint). Today you see that much as before the USSR, most Russians see the church as a second arm of keeping life stable and providing answers, a cultural side to the state's function.

To summarize: USSR suppressed religion but never to the point that main institutions were eradicated. Religion became popular in the aftermath of its fall as Russians needed structure in life.

The Soviet Union did not really eliminate religion. Rather, it eliminated religion that refused to be controlled (or was suspected of such) by the State. From the first book of Solzhenitsyn's Gulag Archipelago, we can see how Soviet authorities essentially purged the leadership of the Orthodox Church and any priests or prominent lay people who refused to join in on the orgy of denunciation and condemnation. The Soviet government also commandeered massive amounts of Church property, often taking local churches and turning them into motor pools, storage warehouses, or, in the case of the largest cathedral in Moscow, the Church of Christ the Saviour, into a public swimming pool.

It should be noted, however, that the Church was not gone from Soviet society. Much of the Church was brought under the control of the security services (OGPU, NKVD, KGB), and it was used as another means of sniffing out dissidents and ministering Soviet authority. A very memorable example of the Church being used as an agent of the State comes during the opening days of Operation Barbarossa, when the Patriarch of the Church delivered a rousing message to the people of the Soviet Union calling them to arms for the defence of Holy Russia against the invading Nazis (which I can't seem to find on Google at the moment).

Will return to this comment later on how religion made (or didn't make) such a prominent comeback.