What are the main differences between tubers and taproots?
They are actually completely different plant parts: a taproot is the central root of a taproot-system plant while a tuber is an engorged modified underground stem called a rhizome. 


A morphological and functional difference between rhizomes and roots are that rhizomes (being modified stems) have nodes, while roots never do. 


Note: a node = the location on a plant stem where buds (leaves, flowers, stem branches) initiate



According to "Plant Identification Terminology: An Illustrated Glossary" (Harris &amp; Harris, 2001)1:


  Taproot. The main root axis from which smaller root branches arise; a root system with a main root axis and smaller branches...Figure 1273




  Root. The portion of the plant axis lacking nodes and leaves and usually found below ground.



Compared to...


  Tuber. The thickened portion of a rhizome bearing nodes and buds; underground stem modified for food storage. Figure 1366




  Rhizome. A horizontal underground stem. Figure 1052



 

 

See here for a photo of a potato plant showing a young tuber and an older tuber growing on the end of a rhizome.

According to here:


  Tubers-are swollen regions of stems that store food for subsequent growth. The potato is an example. It is a stem because it has many nodes called eyes with spaces between eyes known as internodes. Potato tubers develop at the end of swollen underground stem structures, rhizomes.
  
  Eyes of potatoes are really axillary buds which contain several small buds at each site. These buds can expand to form shoots which grow on to make whole plants.


You can explore histologic, anatomic, and physiologic differences of stems vs roots more generally to understand broader differences between the two plant parts. 


Examples: 


Proença &amp; Sajo (2008)2 discuss root and rhizome anatomy in select bromeliads.
Figure 1 from Hernán et al. (2014) shows how the vascular and various tissue layers can differ quite a bit between roots and rhizomes on the same plant. 



Notes:


Some structures characteristics of roots (e.g., casparian strips) can also occur in rhizomes. (E.g., see Lersten 19974.
Unlike potatoes (which are tubers), sweet potatoes (Ipomoea batatas) are actually a modified root called tuberous roots




Works Cited:

1 Harris, J.G. and Harris, M.W., 1994. Plant identification terminology: an illustrated glossary (No. QK9 H37 2001). Utah: Spring Lake Publishing. 

2 Proença, S.L. and Sajo, M.D.G., 2008. Rhizome and root anatomy of 14 species of Bromeliaceae. Rodriguésia, 59(1), pp.113-128. 

3 Hernán, G., Varela, B.G., Fortunato, R.H. and Wagner, M.L., 2014. Pharmacobotany of two Valeriana species (Valerianaceae) of Argentinian Patagonia known as “Ñancolahuen”. Lat Am J Pharm, 33(6), pp.891-6. 

4 Lersten, N.R., 1997. Occurrence of endodermis with a casparian strip in stem and leaf. The Botanical Review, 63(3), pp.265-272.