I believe there were several weaknesses that Beethoven carried with him through life. One was the loss of hearing that began to onset when he was in his twenties. Although he continued to compose music for the next thirty years, it severely impacted his interactions with those who could have become larger patrons and other social activities where he might have found some enjoyment. Instead, he kept to himself, except from some of his closest friends, whom he trusted explicitly.
The other would have been his temper. He had a terrible temper, and it bested him a few times when it could have benefitted him if he had managed to keep his temper in check. However bad his temper got, there were a few patrons that continued to support him no matter what. His circle of friends was small, but they were loyal, and he rewarded that friendship with some beautiful melodies.
The final weakness was the depression that he suffered from. You can pick up on some of it from his later works. They are darker, more emotional, and the music feels more unstable than some of his other works.