Short version:

Barrister = a gentleman, typically a younger son of an aristocrat or a member of landed gentry, who acts as a trial lawyer and point of law within his legal expertise. Considered a gentleman because he does not receive a salary, does not apprentice, does not work with clients directly, rather he receives a gift or honorarium from the solicitor for agreeing to represent the case in court.

Solicitor = a working man who aids clients with legal paperwork, ie contracts, wills, deeds, etc. Not considered a gentleman, as he receives a salary.

Not only do they fulfill vastly different roles, their education, training, earnings, and governing bodies differ. If wishing to learn more about each, definitely look into their education requirements. A solicitor, for example, would apprentice as his training rather than attend university. A barrister, however, would be a university educated man who then joined one of the Inns of Courts to study with the barristers of that Inn before being called to the Bar. Both had strict requirements before entering their profession (barrister) or trade (solicitor), so give those a stern eye to fully understand the difference between the two. Even the language requirements would have been different, for much of the court cases would have been conducted in a combination of Latin, French, and English, more so with Latin earlier in the 18th century, but by late 18th century, it had switched to English with the lingering combination of "law Latin" and "law French."

If our Georgian era hero is a gentleman, he would be a barrister. If our Georgian era heroine needed legal aid, she would seek a solicitor, who could then present the case (if applicable) to a barrister if they needed legal representation in court. If our Georgian era hero is a working man rather than a gentleman, then he would likely be a solicitor.

Could a working man become a barrister in the Georgian era? Unlikely. With a great deal of money, experience, and elbow rubbing, it was possible, but unlikely. Would a gentleman become a solicitor in the Georgian era? Even less likely. Anything is within the realm of the possible, but not likely.

A few great posts to learn a little more:

Golden Romance Research: University Life

Regina Jeffers: Barristers and Solicitors

Random Bits of Fascination: The Gentlemanly Professions