The most obvious effect is the lack of a suitable rocket to lift heavy cargo into orbit. That set a constraint, especially on the manned space program. For example, the N1 was basically the backbone of the Soviet effort for a manned mission to the Moon, and a termination of the program meant that the most basic requirement for that kind of mission was gone. Similarly, even more distant goals, like a Mars mission, which have had some possible mission architectures sketched,needed a N1 or a rocket with similar payload capacity in order to be possible at all.
The direction the Soviet space program went instead was therefore stand alone, and later modular, space stations in low Earth orbit, with interplanetary missions limited to small robotic spacecraft. That is pretty much how space travel is today, and how it has been for approximately 40 years both in Russia and in the US. The brief appearance of a new heavy lift launcher, Energija, did not change that.