They have the same meaning but a different register because of the different ending forms.
There are several ending forms in Korean depending on the exact meaning the speaker wants to express.
Learners first need to know the four standard ones of these with no special connotation other than formality and politeness.
Formality is about how official(stiff) or familiar(soft) it sounds, and politeness how respectful one is to the listener.
Here are the four common declarative endings in the present tense, with the 피곤하다 example.
1. -ㅂ니다 (피곤합니다) - [formal, polite] For someone older than you or a group of people in a formal setting.
2. -다 (피곤하다) - [formal, plain] For addressing the public through a media, or when speaking to friends.
3. -아요/어요 (피곤해요) - [informal, polite] For adults outside of your close circle.
4. -아/어 (피곤해) - [informal, plain] For friends and siblings, or children.
* These four styles are not strictly only for the occasions listed above - we often mix them in ordinary speech.
So both 피곤해요 and 피곤하다 are natural and in common use, but usually to different types of people.