In 1961, after several years of enthusiasm about 'cybernetics' and computers as 'machines of communism', a group of Soviet scientists made a detailed proposal of a series of computer centres across the USSR that would be networked together to share data. The Communist Party under Khruschchev initially thought this was a fine idea indeed: their 1961 Twenty-Second Congress Of The Communist Party included in its manifesto the idea that cybernetics "will be widely applied in production processes in manufacturing, the construction industry and transport, in scientific research, in planning and designing, and in accounting and management.’" This was the same year that Yuri Gargarin went up in space, and communists were confident that the USSR could manage such technology very well indeed. The West was also concerned by this at th etime - one American reviewing this proposal in 1963 thought that this technology might be what Khrushchev thought would 'bury the West'.
However, these grandiose plans never quite came to fruition, very often because of Soviet political considerations overriding plans to develop the technology. The people in charge of things like accounting and management strongly opposed computerising these systems, for example, because it would expose their inefficiency and/or upset the balance of power. And the plans would only work if fully realised, and the time it would take to do this was another issue - as the Soviet power structure evolved, the extent to which it was receptive to such changes also evolved. So a formal plan presented in 1964 based on the ideas of 1961 was no longer being presented to Khruschchev. Instead, by this point, Brezhnev was in power. The formal plan was rejected.
However, from the late 1960s, a network of eight computers was put together to co-ordinate the A-35 missile system and to serve as an early warning system. There was another separate network which also partially served as an early warning system for Moscow and Riga, and a third separate network designed to organise the Strategic Missile Forces. In this era, different Soviet ministries and even different parts of the military did not communicate with each other. Many things were carried out in secrecy, with limited collaboration between different departments. As computers developed, there were something like 300 different computer systems in the USSR, built by different teams for different purposes, with little knowledge of each other and little incentive to collaborate. In the 1970s, each ministry had its own computer network, but there was no communication between ministries - and as the ministry systems got more elaborate and complex, communication between different systems became more and more difficult. There was also no civilian access to any of these systems.
Soviet awareness of ARPANET in the late 1960s did change things. Another (even more ambitious) proposal for an intergrated Soviet computer network in 1971 was passed in a draft resolution of the Twenty Fourth Party Congress before being quickly retracted after the powers-that-be realised that it threatened to upset the established balance of power. In the 1980s, AKADEMSET came into existence for the use of academics, but it was very limited; the UN report suggests strong evidence that it never worked at all. A second network for the use of academics, IASnet, designed to give Soviet academics access to international databases, appears to have been more successful. However, it was also heavily controlled - requests for IASnet access had to go through bureaucratic gatekeepers, and it was widely seen as not worth the effort.
Towards the end of the 1980s, private, unofficial internet services started to make a mark, but they were clearly based on pre-existing American infrastructure rather than innovative. Fidonet bulletin board systems were popular amongst computerphiles in the USSR and often served the purposes that AKADEMSET was supposed to. And a computer network that was started as part of a Russian equivalent to UNIX in the late 1980s, Relcom, began to be popular, and - when the Soviet state collapsed, became a pay-as-you-go service.
So in terms of breakthroughs and innovations, the Soviets were generally behind the times, thanks to political considerations stymieing ambitious plans in the 1960s and 1970s. And where breakthroughs and innovations might have been made in individual systems in individual ministries, there was no infrastructure for those breakthroughs to be translated to wider use. The culture of secrecy and the lack of collaboration between different ministries (as well as the general inability of private enterprise to effectively fill the gap in the controlled Soviet economy) meant that innovation and breakthroughs would not have been known about and thus wouldn't have promulgated through subsequent projects across the economy.
Sources:
a 1999 UN report called Mapping Russian Cyberspace: Perspectives On Democracy And The Net
Slava Gerovitch's 2008 paper in History And Technology, 'InterNyet: Why The Soviet Union Did Not Build A Nationwide Computer Network'.