The terms "bonsai" and "topiary'  both refers to plants that are styled by pruning but they are produced by different techniques.

Bonsai

Bonsai is a Japanese word meaning "tray planting". It is often used to mean miniature trees grown in containers but this is not strictly correct since bonsai do not depend on dwarf varieties but on the use of pruning and manipulation of the tree to keep it small.

Essentially any tree can be grown as a bonsai, but some species are preferred over others. The size of the tree is restricted by the small pot in which it is grown. Its small size is further restricted by leaf trimming, trunk and branch trimming and also root trimming. Bonsai are often trained into interesting shapes by wiring and using mechanical means to shape the tree. 

Topiary

This term refers to perennial plants (usually trees or shrubs) that have been pruned into particular shapes. They may be regarded as "living sculptures". Unlike bonsai, they are usually large and grown directly in the garden in plots or in lawns (although they may also be grown in pots). 

Topiaries can assume many forms. Some may be strictly geometrical (e.g. cubes or spheres) whereas others can assume fanciful shapes (e.g. animals).  In this way they differ from bonsai which often assume the shape of the native species, although frequently in a convoluted form. Topiaries are usually produced from evergreen plants with small leaves or needles, such as boxwood (Buxus semperviriens), yew (Taxus spp.), arborvitae (Thuja spp.) and holly (Ilex spp.). 

Courtesy of NYBG Plant Information Service