I find "functional PCA" an unnecessarily confusing notion. It is not a separate thing at all, it is standard PCA applied to time series.
FPCA refers to the situations when each of the $n$ observations is a time series (i.e. a "function") observed at $t$ time points, so that the whole data matrix is of $n \times t$ size. Usually $t\gg n$, e.g. one can have $20$ time series sampled at $1000$ time points each. The point of the analysis is to find several "eigen-time-series" (also of length $t$), i.e. eigenvectors of the covariance matrix, that would describe "typical" shape of the observed time series.
One can definitely apply the standard PCA here. Apparently, in your quote the author is concerned that the resulting eigen-time-series will be too noisy. This can happen indeed! Two obvious ways to deal with that would be (a) to smooth the resulting eigen-time-series after PCA, or (b) to smooth the original time series before doing PCA.
A less obvious, more fancy, but almost equivalent approach, is to approximate each original time series with $k$ basis functions, effectively reducing the dimensionality from $t$ to $k$. Then one can perform PCA and obtain the eigen-time-series approximated by the same basis functions. This is what one usually sees in the FPCA tutorials. One would usually use smooth basis functions (Gaussians, or Fourier components), so as far as I can see this is essentially equivalent to the brain-dead simple option (b) above.
Tutorials on FPCA usually go into lengthy discussions of how to generalize PCA to the functional spaces of infinite dimensionality, but the practical relevance of that is totally beyond me, as in practice the functional data are always discretized to begin with.
Here is an illustration taken from Ramsay and Silverman "Functional Data Analysis" textbook, which seems to be the definitive monograph on "functional data analysis" including FPCA:
One can see that doing PCA on the "discretized data" (points) yields practically the same thing as doing FPCA on corresponding functions in Fourier basis (lines). Of course one could first do the discrete PCA and then fit a function in the same Fourier basis; it would yield more or less the same result.
PS. In this example $t=12$ which is a small number with $n>t$. Perhaps what the authors view as "functional PCA" in this case should result in a "function", i.e. "smooth curve", as opposed to 12 separate points. But this can be trivially approached by interpolating and then smoothing the resulting eigen-time-series. Again, it seems that "functional PCA" is not a separate thing, it is just an application of PCA.