"The Internet is a global system of interconnected computer networks that use the Internet protocol suite (TCP/IP) to link several billion devices worldwide." [wikipedia] The term Internet, with capital I, refers to this specific network of interconnected devices. Internet is part of the Third Industrial Revolution.
Internet of Things (IoT) is a more general concept and encompasses not only Things like computers but also all other aspects of connecting and using all sorts of sensors and controls. In essence, IoT is the Fourth Industrial Revolution.
Figure: 1st-4th Industrial Revolutions

Graphic source from SOGETI (2014)d
Looking at the web-based part of Internet, it started in the early 1990s and it took about 25 years to reach 1 billion web sites. Various estimates for IoT indicate 25-50 billion devices connected directly and indirectly to the Internet in the next 5 years.
Figure: Web 0, 1.0, 2.0, and IoT

Graphic source from SOGETI (2014)

Graphic by Iot Analytics (2014)
The figure below is a good exemplification of aspects included in IoT, also known as Internet of Everything.
Figure: Internet of Things (IoT) / Internet of Everything (IoE)

Graphic: How the Internet of Everything... Cisco 2012
See also