When we speak about modern and contemporary art, the first major difference is related to – time. Modern art was an artistic era which lasted from the 1860s up to the 1970s, whereas contemporary art came AFTER modern art, i.e. in the 1970s, and it lasts up until today. That is the first major distinction between the two eras.
The second difference is in the philosophy behind the two movements. Modernism was all about rejecting traditional artistic conventions which then resulted in creating new movements, such as Fauvism, Cubism, Futurism and Expressionism (among many others). However, contemporary art is yet another concept.
The turning point
According to Art and Culture, a turning point between modern art and contemporary art was the movement called Abstract expressionism.  It was a new, abstract style where the artist just drips or pours paint on the canvas, and the focus lays on the end result which is supposed to convey strong emotional messages. According to Anirudh, the main proponent of this movement was the painter Jackson Pollock, who also became known as Jack the Dripper, because of his own particular style of painting. 
This was also the beginning of contemporary art, which was characterized by a high degree of abstraction, as well as by questionable aesthetics. This means that the observer of the art work of this period, by looking at the work of art, would question him or herself if this really is art or not, because the belief was present that even a child could make these paintings. This is the goal which contemporary artists have – to make the public question itself whether something is art or not.
 Art and Culture, Internet: https://artsandculture.google.com/theme/vwKiW17vbvl3JA [Accessed 19-12-2018]
 Anirudh, 10 Most Famous Modern Art Artists and Their Masterpieces. 2017. Internet: https://learnodo-newtonic.com/famous-modern-painters [Accessed 19-12-2018]