The differences
There are temps agencies that specialize in a particular industry, such as construction, food or hospitality. There are also offices that specialize in types of jobs, such as technical jobs, or jobs in IT. Sometimes you can tell from the name, but not always. Want to be sure? Then look at the office's job listings, or just ask.
Employment agencies do not have the same clients or job openings. If you're going to apply to temp agency A, you're applying for a job with the companies that temp agency A works with. Employment agency B works with other companies, and so may offer you different job opportunities.
Consultants are people, and they differ, too. So they may handle a job interview differently or you may get along with one consultant and not another.
The similarities
What temp agencies do have in common is that they look for people to work for their clients. They meet and interview candidates. Then they look for a suitable job for those candidates.
Temporary employment agencies also take care of the administration surrounding an employment. That's the contract, insurance, wages, and so on.
Do you have to apply to different employment agencies?
That's up to you. Do you have a good feeling about a temp agency, and do you think they totally understand what you want? Then you don't need to apply to another office. But do you want to increase your chances of finding a new job? Then it's best to visit several offices.