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Among the list of tasks in machine learning, synthesis and sampling is one of the key task. Consider the following explanation regarding synthesis and sampling task from Chapter 5: Machine Learning Basics from the book titled Deep Learning (by Aaron Courville et al.)

In this type of task, the machine learning algorithm is asked to generate new examples that are similar to those in the training data. Synthesis and sampling via machine learning can be useful for media applications when generating large volumes of content by hand would be expensive, boring, or require too much time. For example, video games can automatically generate textures for large objects or landscapes, rather than requiring an artist to manually label each pixel (Luo et al., 2013). In some cases, we want the sampling or synthesis procedure to generate a specific kind of output given the input. For example, in a speech synthesis task, we provide a written sentence and ask the program to emit an audio waveform containing a spoken version of that sentence. This is a kind of structured output task, but with the added qualification that there is no single correct output for each input, and we explicitly desire a large amount of variation in the output, in order for the output to seem more natural and realistic.

The explanation does not mention any difference between the two tasks. Both sampling and synthesis, apart from the linguistic differences, I don't know any discriminating criteria, qualities or properties, that separate both tasks in machine learning.

What is the fundamental difference between sampling task and synthesis task in machine learning?

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The terminologies can be confusing because of the different ways authors use them. The bottom line is this

The Synthesis task basically refers to creating or synthesizing new data. Creation of data can be purely deterministic, e.g.

.. we provide a written sentence .. to emit an audio waveform containing a spoken version of that sentence

But, a vast majority of the time, such creation has an ..

.. added qualification that there is no single correct output for each input, and we explicitly desire a large amount of variation

which is done by a Statistical Sampling procedure. This procedure can be as simple as IID sampling from $\mathcal{N}(\mu, \sigma)$ to more complex MCMC methods.

So basically, "Synthesis" is a broader task that may contain a "Sampling" procedure.

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