Well the biggest difference is your manager most likely works for your band and your band only- or at most a very small circle of acts. Managing a good act is pretty much a full-time job. Your manager is normally your proxy and will have authority to sign contracts on your behalf. Their allegiance should be to YOU.
An agent is an independent company/individual who works with dozens if not hundreds of acts - not just you - and might also be working for the venues and not the bands. Their job is to put the band and the venue together on a certain night or range of nights. The agent will normally not get involved with the specifics the band or the club needs (contract riders) - that's where management comes in.
A manager usually handles your money - they're the third-party who can make or break a band financially. They are responsible for getting the money from the venue, paying the booking agent, the equipment rental, the lighting company, the trucking company etc. Depending on the size of the act, the financial manager might also act as a road manager, making sure everything gets to the right place at the right time.... and even a personal manager... making sure you and your other musicians are taken care of in terms of running errands, taking care of incidentals etc. Of course with larger acts "Management" could be several individuals, one dealing with money, one with gear, etc etc. The manager is also usually the contract proxy for the band - they are authorized to negotiate and sign contracts for everything from playing to recording and they also handle the contract riders --- how many dressing rooms... what's in the dressing rooms.... how the stage is to be set up....etc. All that stuff. Many a band has been bankrupted by an inept or crooked manager.
Booking agents are independent companies/individuals who work with dozens of acts and clubs. When I was playing full-time we had 2-3 booking agents working for us in various parts of the country, and each had a couple hundred bands and probably just as many clubs they did business with. A booking agent finds and connects the band with the gig - but they do not normally have final contract authority. The band and their manager still needs to be involved to OK the deal (or not) and deal with enforcing the contract, although the agent certainly has a role in that as well. Note that booking agencies are not always protagonists for the band - they could be retained by the CLUB to find and book the entertainment for that club. In that case the band would be foolish not to have their own representation since the Agent is actually working for (and being paid by ) the venue not the band.
As with anything else in life - there are plenty of gray areas here. Booking agents who also manage acts..... Managers who do all their own booking with clubs... etc. There is no "one size fits all".
I hope that helps.