Question: There are a lot of references in the statistic literature to "**functional data**" (i.e. data that are curves), and in parallel, to "**high dimensional data**" (i.e. when data are high dimensional vectors). My question is about the difference between the two type of data. 

When talking about applied statistic methodologies that apply in case 1 can be understood as a rephrasing of methodologies from case 2 through a projection into a finite dimensional subspace of a space of functions, it can be polynomes, splines, wavelet, Fourier, .... and will translate the functional problem into a finite dimensional vectorial problem (since in applied mathematic everything comes to be finite at some point). 

**My question is:** can we say that any statistical procedure that applies to functional data can also be applied (almost directly) to high dimension data and that any procedure dedicated to high dimensional data can be (almost directly) applied to functional data ? 

If the answer is no, can you illustrate ?

EDIT/UPDATE with the help of Simon Byrne's answer:

* sparsity (S-sparse assumption, $l^p$ ball and weak $l^p$ ball for $p<1$) is used as a structural assumption in high dimensional statistical analysis.
* "smoothness" is used as a structural assumption in functional data analysis.

On the other hand, inverse Fourier transform and inverse wavelet transform are transforming sparcity into smoothness, and smoothness is transformed into sparcity by wavelet and fourier transform. This make the critical difference mentionned by Simon not so critical?

Answer: 
Functional Data often involves different question. I've been reading Functional Data Analysis, Ramsey and Silverman, and they spend a lot of times discussing curve registration, warping functions, and estimating derivatives of curves. These tend to be very different questions than those asked by people interested in studying high-dimensional data.
Yes and no. At the theoretical level, both cases can use similar techniques and frameworks (an excellent example being Gaussian process regression).

The critical difference is the assumptions used to prevent overfitting (regularization):

* In the functional case, there is usually some assumption of smoothness, in other words, values occurring close to each other should be similar in some systematic way. This leads to the use of techniques such as splines, loess, Gaussian processes, etc.
* In the high-dimensional case, there is usually an assumption of sparsity: that is, only a subset of the dimensions will have any signal. This leads to techniques aiming at identifying those dimensions (Lasso, LARS, slab-and-spike priors, etc.)

UPDATE:

I didn't really think about wavelet/Fourier methods, but yes, the thresholding techniques used for such methods are aiming for sparsity in the projected space. Conversely, some high-dimensional techniques assume a projection on to a lower-dimensional manifold (e.g. principal component analysis), which is a type of smoothness assumption.