Discrete vs Continuous Signal
A continuous time signal can be evaluated arbitrarily at any time,at least in theory, and can take on values at arbitrary points between any evaluation point. The Fourier transform, while not always well defined, can take on values at frequencies over $-\infty$ to $\infty$ and that value can be zero, and the signal in continuous time also takes on values over the infinite interval and those values can also be zero.

A discrete time signal only takes on values at discrete time intervals. It is only meaningful to consider values intermediate in time, intermediate to those  samples,  if for example the discrete time signal was sampled from a continuous signal, or if you have a reason (justification). In general, the values intermediate to the samples are not defined. You can consider the price of a stock or pork bellies on days where there are no trades. No one is buying or selling. The price isn’t the last traded price and it isn’t zero. It does have a Fourier Transform, computed as an infinite summation, and the frequency is uniquely (continuously defined) limited over $-\pi$ to $\pi$.

So there is only a subset of discrete time signals that we can meaningfully compare their Fourier Transforms, and those are signals that are Nyqvist Sampled.  Band limiting and periodic sampling alter the original continuous time signal, which alters the frequency spectrum, which almost answers your question.  

The question of whether something is meaningful at intermediate points between samples in both time and frequency is more subtle. 

I can have a periodic discrete time sequence and the frequencies in the Fourier transform are likewise values that are discrete in frequency, but are the intermediate values in frequency zero?

The most used DSP text book is  Oppenheim and Schaefer and the most recent edition has 3 interpretations of the DFT which I interpret as yes , no, and maybe. 

I can also interpolate pork belly prices, but is it meaningful?

I think the answer lays with how you want to represent something. 

If you want to describe a discrete time signal in the continuous time domain, a sequence of Dirac Delta functions is suitable where intermediate values are zero is OK. In other contexts, it may be meaningless.  

Another example worth considering is a realization of a Poisson process.  You have a set of "arrival times" that you can store on a computer, limited by the amount of memory and numerical precision available on the machine. It is a list of numbers and the intermediate values denote no "arrivals" at those intermediate times.  The Poisson process is continuous time.  One typically doesn't perform Fourier analysis on a Poisson Process. There is some discretization in the computer representation, so one can quibble that it isn't strictly continuous but it is a practical representation of a continuous signal.  This is also an example of a continuous signal where Nyquist sampling is irrelevant.