Adding to what Shashi said, they have different possibilities and use cases.
MQTT defines a standard wired protocol for pub / sub and, as Shashi noted, is designed for very light environments. Thus, it has a very minimal wire format, several basic qualities of service and a basic set of functions.
Traditional message queuing systems, on the other hand, are usually proprietary (although AMQP seeks to change this), cover both point-to-point and pub / sub, offer many quality of service and tend to have a heavier wire format, although this exists to support advanced feature sets, such as addressing response, protocol translation, etc.
A good example of MQTT would be where you have endpoints in phones, tablets, and set-top boxes. They have minimal power, memory and system resources. As a rule, the connections from them either remain MQTT, and they talk to each other, or they connect to enterprise-class MQ, where they can interact with internal applications. For example, an MQTT-based chat client can talk directly to another through an MQTT broker. Alternatively, an MQTT-based content delivery system could connect to a corporate messaging network that hosted ads and other content to be delivered to applications running on phones and tablets. The internal base of the enterprise will manage all the statistics of the delivery of advertising and the views on which the accounts are based, and the MQTT foot allows you to download content with minimal battery or power consumption on the end device.
Thus, MQTT is used for embedded systems and end-user devices in which problems with power, bandwidth and network stability. This is often combined with traditional MQ messaging, although I have never seen MQTT used as an exclusive transport for traditional messaging applications. Presumably, this is due to the fact that MQTT does not have some more robust features, such as message correlation, reply to addressing, and point-to-point addressing, which have been the basis for messaging for 20 years.