Varieties of criticism
Conservative criticism has nothing much to do with "left-wing" or "right-wing", because left-wing people are often very conservative, in the defined sense, while right-wing people can also be very radical, in the defined sense. The difference between "radical" and "conservative" has more to do with the belief in whether "a change to something genuinely new" is really possible and necessary. Radicals typically believe strongly that such change is highly desirable and necessary, and that it can be achieved. Their criticism is that there is "not enough" change. Conservative critics, by contrast, are very skeptical about any such change, because they feel the change is really "just another form" of something that already exists.
Right-wing authoritarianism
According to Karen Stenner, an Australian professor who specializes in authoritarianism, says it is different from conservatism because authoritarianism reflects aversion to difference across space (i.e. diversity of people and beliefs at any given moment), while conservatism reflects aversion to difference over time (i.e. change). Stenner argues that conservatives will embrace racial diversity, civil liberties and moral freedom to the extent they are already institutionalized authoritatively-supported traditions and are therefore supportive of social stability. Conservatives tend to be drawn to authoritarianism when public opinion is fractious and there is a loss of confidence in public institutions, but in general they value stability and certainty over increased uniformity. However, Stenner says that authoritarians also want difference restricted even when so doing would require significant social change and instability.