"Gladness" is more a reaction you have to something, whereas "happiness" can exist all on its own. You can be happy for no particular reason, but if you say you feel "glad", people are likely to ask what it is that you are glad about. The words do overlap though. You can be happy about something in particular.

They mean the same thing but happy is more commonly used. In everyday use we say “I’m feeling happy today” more often than “I’m feeling glad today” It’s common to hear “I’m glad to see you” or “I’m happy to see you”. If you are not sure which to use in conversation “happy” is the safest choice.