As far as importance in engineering is concerned ,it depends on interest .
for example,if you are doing studies in robotics it would help you to create a finite state automata that controls states of your robot.
One of the most important implementation of Automata is Compiler Designing !
If there are no practical exposure then those subjects will restrict to only "marks" for your course.
We can find different levels of complexities in every field which needs scientific understanding and analysis.
Stephan Wolfram who is a leader in automata applications states that entire universe has set of states and rules and single condition.
Automatas are applied in fields including :
- Fluid Flow
- Snowflake and crystal formation
- Chaos theory
- Financial analysis