Apache Tomcat is a no brainer to run if you are a Java shop. It meets all the Java 7 and Java 8 web runtime needs without a bloated JEE overhead. It works great for Spring, Spring Boot, Dropwizard and other modern microservice frameworks. It is speedy enough and runs with minimal or no setup.
It is not quite as fast as Jetty, I sometimes use Jetty for Spring Boot services sometimes Tomcat. They interchange easily. Java applications with JDBC to a SQL database are probably better off with Tomcat. Tomcat also has a ton of versions so it's a bit of pain to manage all that you may have in your enterprise. If you are doing traditional WARs then you have a lot of configuration files to touch and you have to make sure you have your SQL configs correct. A Paas is easier for that.
Try it it's a quick and easy download. Or you can just use it through Spring Boot, it's the default built in web run agent. It works easy for WARs or JARS through Spring Boot.
Running web applications for free with open source. We have achieved that. Performance and stability at zero cost. hard to argue with that success. Runs all our JRuby, Java, Scala, Groovy and other JVM web apps with no issues. We also use it with some open source web applications and they run without admin intervention.