One of the biggest problems Java faces for real adoption amongst tech people is still the perception (and I am not sure it is entirely flawed either) that it is insecure. Yes Flash is just as bad if not worse, but we are stuck with them because they are so ingrained within the Internet. Java never caught on, and I know that when I see a Java application I let out a huge sigh and go download and install Java again. I'…
The biggest problem Java Applets had and still have is that it's so much pain to update the Java plugin that most users are left with old versions that have documented exploits. I don't get why it doesn't auto-update itself - if browsers like Chrome or Firefox can auto-update themselves automatically, why can't the Java plugin?
It's better that they don't. There are often subtle differences between versions. Sometimes it causes some JDBC drivers to fail. Sometimes it can cripple entire platforms like 1.7 did with Solr for instance.