The information is exactly the same but the sentences "feel" different. The strongest of the three sentences is the last one: "I feel happy". It uses an active verb, "feel", not merely a participle, that actually says something. The only verb in the other two sentences is "am". "Am" doesn't actually say anything. It only links adjectives to the subject of the sentences. The second best of the three sentences is the middle one: "I am feeling happy". Both adjectives "feeling" and "happy" describe the subject of the sentence, so the message is strong and clear. The third best is the first one. All it does is to use the linking verb "am" to tie the single adjective "happy" to the subject "I". Without any mention of "feeling", the happiness is portrayed with less strength.

They have practically the same general meaning. But there are slight differences in nuance. "I'm happy" is very simple and generic, and focuses on the 'happy' part. "I'm feeling happy" also emphasises 'feelings', so if you don't want to appear emotional and "in your feelings" you might want to avoid this one. "I feel happy" is somewhere between the two. Also note that "I'm feeling happy" is more of a temporary, short-term thing, often followed by "right now". Meanwhile, "I feel happy" and "I'm happy" can comfortably refer to both short-term and long-term feelings, so these are the better choices for talking about if you're happy with your job, where you are in your career, and how your marriage is getting on, etc.