Forgot about this one, sorry... Here's a quick off-the-cuff guess:
Network analyzers are used to measure the frequency response of networks. They do not intrinsically operate in the time domain, but many include an inverse-FFT function that provides a calculated time response to a step or impulse input. Think of a box with two ports, but you can't see inside. You apply a signal to one port and measure at the other, and repeat in the other direction (most do the switching automatically to measure both ports). The network analyzer measures the frequency response, the transfer function, from one port to the other by sweeping the signal frequency at one port and measuring the resulting signal at the other. From that you may infer some of the properties of the box. Thus, a network analyzer includes a source (sweep generator) and receiver that are synchronized to provide the response at each frequency point in the sweep. Simple ones measure the magnitude only; more complex models measure magnitude and phase and thus provide vector information.
A spectrum analyzer measures the frequency spectrum of an input signal and is what one uses to determine discrete frequency components in a signal. A spectrum analyzer does not include a source. It can be used to measure the signal-to-noise ratio (SNR) by summing all the frequency components except the fundamental (test tone) and comparing the magnitude of the sum to the magnitude of the test tone. Similarly, it is used to provide the distortion (THD, IMD) of a test tone by comparing the magnitude of the fundamental to the sum of all the distortion components. (Not that the “sum” is actually a root-sum-square, the square root of the sum of all tones squared.) You could make a network analyzer by synchronizing a sweep generator and a spectrum analyzer.
A signal analyzer is generally a multi-function device that measures frequency response, SNR, THD, IMD, etc. Most audio test instruments are multi-purpose devices, including a signal source to provide fixed or swept tones and a sensitive receiver that can serve in a network or spectrum analyzer role.
Dynamic analyzers allow you to measure signals over time, allowing you to perform for example a plot of the decay in frequencies over time in a room, or plots of distortion vs. power over multiple sweeps.
The output of a network analyzer is a frequency sweep like this:
The output of a spectrum analyzer looks like this: