With respect to your first question, a patent attorney is a patent agent who is also an attorney. To answer your last question, a patent agent who eventually becomes an attorney is thereafter a patent attorney without filing any additional papers (to the best of my knowledge).
In practical terms, the difference is that an attorney is licensed to render legal opinions (the hallmark of practicing law). While preparing papers on behalf of clients in the PTO implicitly requires even an agent to state to the examiner that this or that claim comports with this or that law, you should consider this a narrow exception.
Here are things a patent attorney can do that a patent agent can't. Render a legal opinion to a client regarding the validity or infringement of a patent. Dabble in licensing and/or litigation.
Here are things both patent attorneys and patent agents can do. Draft and file patent applications. Prepare and file arguments and amendments in the PTO on behalf of clients. Counsel clients on procedures in the patent office (such as the lack of benefits of provisional applications).
In general, I think it would be a good idea to take the registration examination and become a patent agent while in law school. There are a number of benefits. One is that you get to actually practice law (albeit in a limited way) before being an attorney. That can give you additional insight into the law and the way some things work that you otherwise wouldn't get. Some firms may also give you some credit for years worked as an associate toward partnership. For example, some firms pay associates salaries based on years practicing and offers of partnership come up after 6-7 years. You might end up a year or two ahead of your classmates in terms of salary and partnership track.
However, I think the biggest advantage of all is the 2-3 years' experience you'd gain while working through law school. As I've said before, opportunities for new hires in patent law are limited while attorneys with 3+ years' experience are in relatively high demand (although this demand fluctuates).
I hope that helps.
Regards.