My children were given "First Visit" buttons on their first WDW visit (and my daughter was given a "Happy Birthday" button during a later visit that happened to fall on her birthday) when we checked into our resort hotel. (The celebrations were noted on our reservation - and yes, the buttons are free.) My daughter was given a special cupcake with a candle at Cinderella's Royal Table on her birthday (whether because she was wearing a button or because it was noted on our ADR, I can't say), but that was it. The real charm of the buttons was the lovely greetings the kids got from cast members all over the parks, who congratulated them and said hello. It was a lovely treat, and both children kept those buttons as souvenirs and still have them.
I wouldn't advise anyone to get a button with the expectation of special treatment, or with dishonest intentions (e.g., it's not your birthday but you wear a birthday button anyway -- this isn't directed toward you, OP, but sadly people on these forums have crowed about doing such things in the past) -- but do get any applicable buttons as a souvenir (at your hotel's front desk, or from guest services) and take any positives that come from it (sweet greetings or sweet treats) as an unexpected blessing.