While there are superficial similarities in the Cuban and Soviet national elections, there are critical differences in the role to the Communist Party, the nomination process and even the voting procedures. From Time-Life Books' The Soviet Union (p. 139-40), we have:
The [Soviet electoral] process begins a few months before the voting, with mass meetings to nominate candidates. Typically these meetings are stage-managed so that although three people are nominated for a single seat in the Supreme Soviet, only one of them--the one chosen for the post by the party secretary of the district--can be elected. After a suitable lapse of time, the dummy candidates graciously bow out, leaving only the one candidate. This political gavotte goes on all over the country so that the nominations will produce a balanced, predetermined list of candidates: about a third of the 1,500 candidates will be important party functionaries, government ministers, and top military and police officers; the remainder will provide token representation for various ethnic groups, occupations and even ages. . . .
The voter then faces a ballot paper with one name on it. Dropping that ballot in the box constitutes a "yes" vote. To cast a "no" ballot--writing in another name is forbidden--the voter must strike out the name. However, the voters are not sheep and occasionally refuse to react as expected. If an incompetent party secretary nominates someone who is widely disliked, there may be so many "no" votes that the candidate is defeated. Then a new candidate and a new election (and generally a new party secretary) are called for.
John Dornberg, in his The Soviet Union Today (p. 76), describes the process of voting "no" as follows:
It meant that instead of picking up a ballot and immediately dropping it into the box, such a voter walked past the disapproving stares of all the Party and election officials to one of the voting booths at the far end of the polling place in order to cross out the candidates name.
Although, as I understand it, the degree of Party control varied across the Soviet Union--very strict in Ukraine, much less so in the east and the Baltic republics, for example--it seems even the most fundamental aspect of democratic elections, the secret ballot, was not well established in large parts of the country even in the 1980's. This is not the case in Cuba today where voters may fill out, leave blank or spoil their ballots in the privacy of the voting booth. No one else will know how an individual votes.
In Cuba, candidates for delegates to the National Assembly are nominated not the Communist Party, but by the democratically (and competitively) elected Municipal Assemblies in consultation with the various mass organizations (unions, and farmers', students' and women's organizations). The Communist Party of Cuba, by law, is not even allowed to nominate or endorse any individual candidates.
In 1974, Cuba began to experiment with electoral democracy at the municipal level. By 1976, in Cuba, there were nation-wide democratic and competitive municipal elections. The Municipal Assemblies, at that time, elected delegates to the Provincial and National Assemblies.
China, it should be noted, has only as late as 1998 began to dabble in a limited way in electoral democracy at the village level. These very tentative steps were lauded at the time by US president Clinton and former president Carter, while similar and even more advanced developments in Cuba were completely ignored.
In the late 1980's, there was a great clamor for democratic reforms throughout the socialist bloc, including Cuba. Fidel Castro is said to have cautioned the Soviet leadership about this problem in 1987. They chose to ignore or suppress the problem until, it would seem, it was too late. The resulting collapse, from which most regions of the former USSR have never recovered, has been well documented. This, of course, did not happen in Cuba. There, the leadership responded by initiating a process of genuine, public consultation, the outcome of which was the revised Constitution of 1992. (August, p. 231-3) With these reforms, the role of the Communist Party in elections was further diminished and there were direct elections to all three levels of government--municipal, provincial and national.