I'd like to encrypt some private data by every user and I was wondering if I could use the same secret as passphrase for an RSA private key and as password for authentication. How secure is using an encrypted RSA private key compared to blowfish password hashing?
This is not secure. If you store the key you enable anyone with database access the ability to decrypt the user data.
You should not store the private key. Instead generate it on demand using the password + salt whenever needed. If entering the password every time encryption/decryption is required is not feasible for your use case then cache it in memory and clearing it when the user logs out.
RSA is slow and asymmetric. It also (as pointed out by A. Hersean) doesn't play well with key derivation algorithms. I would suggest you should probably be using a symmetric algorithm like AES for encrypting user data.