On the job, there are at least seven areas that I would list as important ones for helping to advance your career.
Remembering specific tasks
Remembering consumers
Remembering co-workers
Remembering requirements
Remembering essential tools
Remembering trivial items
Remembering feedback
Remembering specific tasks is, ultimately, the main requirement for getting the job done as it should be. Whether you're a perfectionist or not, you can bet that either your boss or your boss' boss is and you will be expected to remember exactly what needs to be done and to have that area completed in a specific amount of time. For instance, if I were delivering newspapers, I would realize that I have a specific amount of time to complete my route and I would need to have all of my papers delivered in the amount of time set up by the previous delivery person. Otherwise I could expect some questionable comments from my boss.
Remembering consumers is nearly as important to nearly any job as remembering the specific task at hand. If I'm a professional writer and I forget to remember that I have READERS (or consumers), who I'm writing for, then I'm apt to write something illegible that would be scrawled, scribbled or impossible to read. Whether I'm writing a note for a secretary or a book for a reader, it's very important to remember who I'm writing for and to use my common sense as the main criteria for writing that piece.
Remembering co-workers would come into a close second place with remembering consumers mainly because you and your co-workers are like a team and who would want a team member who completely ignores their other team member's wants, needs and ideas. It might take a little longer to listen to the suggestion or personal information of someone you're working with but there isn't a doubt that you'll be a much happier employee than you would if you totally ignored your co-workers.
Remembering requirements might mean that you will need to put your personal thoughts on hold for the moment. If you're doing nearly any professional job you're going to need to know and understand the specific requirements for the task at hand. Truck drivers are required to do a safety check of their truck and trailer before leaving. A new to the job truck driver might not remember to go through a mental check of his truck before leaving for the open road. If he doesn't remember, though, it could mean the difference in loosing the entire load and the job itself or making the long haul across country without incident.
Remembering essential tools might mean you have an entire truck filled with basic carpentry tools or it could mean that you have a briefcase filled with notes and reminders. Either way, it's very important to your job that you always remember to have those tools on hand because without them, you'll end up spending a fair amount of your time (clocked) in seeking out the tools you do need to complete the task.
Remembering trivial items like a specific customer's favorite restaurant or even their favorite color could have a huge impact on your career. And, even more importantly, remembering the customer's situation and personality can be beneficial to keeping the customer for a long time. If I don't remember that my customer is going through a divorce and happen to ask how his wife is after he's specifically asked me to hold off on the subject, then I've made a big boo-boo.
Remembering feedback is just as important as these other elements for advancing in your career. When someone notices something you've been slack in noticing yourself, remember and change that area. On the other hand, when someone gives you praise for a specific area, remember that too and continue to provide the results hoped for.
I believe a good memory is helpful, not only as a means of advancing in your career, but in your entire life as a whole, emotionally, mentally, physically and spiritually. Remembering that a particular task needs to be completed at work is essential, of course, to advancing in your career, but there are a number of other areas where a good memory will be helpful, too.