What is the difference between a web developer and a software engineer in web development nowadays?

I'm new to the web development. I've always been wondering what the difference is between a web developer and a software engineer (in web development).

I know that web developer should be familiar with HTML/CSS, and probably a little bit knowledge of Javascript. Meanwhile, software engineer in web development should be familiar with all the things mentioned above, and probably with some database management skills. But is it required that one should be a web developer before proceeding with a software engineer job?

For example, so far I've self-learned the basics of HTML5/CSS3, Javascript, and MySQL. However, I've not yet build a complicated websites such as social networking sites, etc. Will there be any chance that I can apply for a software engineer job in web development and get hired? Or is there a must-do/must-learn before proceeding with this career?

Thank you!