Do all different types of neurons serve different roles?
Short answer
Neurons were (and still are) often identified based on their histological features (e.g., by Cajal). In turn, the structural hallmarks of neurons are often related to their specific functions.

Background
The question is broad and difficult to answer without restricting to some example types of neurons. As you say, about 10,000 types of neurons have been identified. Their roles are often obscure. I will restrict the answer below to a few examples of neurons with relatively well characterized histological features and relatively established functionality.


Fig. 1. Examples of neuronal cell types. A. Purkinje cell B. Granule cell C. Motor neuron D. Tripolar neuron E. Pyramidal neuron F. Chandelier cell G. Spindle neuron H. Stellate cell.source: Ferris (2012) 

The Purkinje cell (Fig. 1A) receives large amounts of input in the cerebellum, integrates it and sends it away to deeper layers. Hence, as an integrator, it has an elaborate input side (large dendritic tree) and a single output axon. The basic  function is similar to the pyramidal cell in the cerebrum (Fig. 1E) and their appearance is hence similar. 

On the other end of the spectrum is the spindle neuron (Von Economo neuron, Fig. 1G), which is believed to receive simple input and generate simple output and is built for speed. Less integration allows for faster responses. Hence, its dendritic tree is much less elaborated than the Purkinje cell (cf. fig. 1A) (Allman et al., 2011). 

As a last example: the Chandelier cell (Fig. 1F); these neurons have axons that form elaborate collaterals that are believed to form local inhibitory connections in the cortex to keep (excessive) stimulation in check; hence their output is more elaborate, with relatively simple input. Dysfunction is believed to be associated to epilepsy (excessive stimulation) (Inan &amp; Anderson, 2014). 

References
- Allman et al., Ann N Y Acad Sci (2012), 1225: 59–71
- Inan &amp; Anderson, Cur Opinion Neurobiol (2014), 26: 142–48
- Purves D et al. (eds.) Neuroscience, 2nd ed. Sunderland (MA): Sinauer Associates (2001)  

Source
- Ferris, 2012, Sci Am blog