In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real projective plane, etc. that have a "discrete" analog: covering graph, graph bundle, discrete Morse theory, abstract simplicial complex, difference equation, finite field, finite projective plane, etc. I would like to know if there are others. But the real question is: Are there any important "continuous" mathematical concepts without "discrete" analog and vice versa?
A lot of ideas from topology and analysis don't have obvious discrete analogues to me. At least, the obvious discrete analogues are vacuous.
The interior of a set.
I think a better question is which ideas have surprisingly interesting discrete analogues, like cohomology or scissors congruence.
Is there a discrete analogue of the notion of discreteness?
A timely example would be the lack of a combinatorial Ricci flow in dimensions $n \geq 3$. In principle I think many people believe there should be one, but a combinatorial/discrete formalism has yet to be found.
The intermediate value theorem wouldn't be true in a discrete setting.
Is "continuous function" an important concept? Does it have a discrete analog?
It seems to me there is no good (powerful) discrete version of Atiyah–Singer theorem.
What do You mean by word analogy here? From wikipedia we have ( among others):
The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a similarity
So You see similarity in differential equation versus difference equation, but this is mostly matter of aesthetic. In practice if You need discrete equation for continues one, You have to put usually a large amount of work in order to make this analogy working. Of course in principle there is relation among differential and difference equation. But what is important here is not what is similar, but what is a gap between them.
When You say, that discrete case may approximate continues one, in fact You take many assumptions, for example about criteria which constitutes what is that mean approximation.
Say what is analogy of holomorphic function? Is discrete complex function on lattice of Gauss integers, good approximation for some complex analytical function? In what meaning? What are criteria? Are all properties of holomorphic function shared by "discrete analogy" and vice versa?
For example, it is not true that whole theory of differential equations may be deduced from difference equations. We have several equations when we cannot find correct approximations, for example Navier-Stokes equation has no discrete model, at least till now. You may say: but chaos is analogous to turbulence. Why? Because is similar? Why do You may say that? Is that someone think two things are similar enough to say that they are?
Then analogy is so broad in meaning word, that I may say, I can see analogy between every things You may point. It may be very useful as inspiration, sometimes it lead us to great discoveries. For every thing You say is analogous to some continues case, we may have differences between them which allows us to distinguish this cases. They nearly almost are non equivalent even in approximate meaning. They are never the same. It is a matter of criteria, if You may say two things are in analogy.
Contrary to the comments appended to the question, I think the notion of analogy can be made precise.
Definition: An analogy of concept A defined in setting SA, is a concept B defined in setting SB such that there exists a generalized setting SX which includes both SA and SB as example settings, and such that there also exists a concept X defined in setting SX which reduces to concept A or concept B when attention is restricted to either setting SA or SB.
In general, an analogy is not unique. A concept could have many analogies, and even for a particular analogous concept there could be more than one way in which it is considered to be analogous.
Example: In Time scale calculus which unifies difference and differential equations, there have been publications with differing answers over how to define the analogy between discrete and continuous transforms. A particular description which encapsulates both the integer and real number transforms may apply to other sets such as the rationals, but a different description might not apply to Q. So an analogy is not just two objects but also the link between them.
Searching Google Web, Google Books and Google Scholar for "no discrete version" OR "no discrete analog" OR "no discrete analogue" OR "no continuous version" OR "no continuous analog" OR "no continuous analogue" produces some examples including a comment that a continuous version of a discrete concept doesn't necessarily enable you to guess the properties of the discrete case.