The answer to that is within the definitions of those two words.

One of the tenets of science is that is empirical. Empiricism means that any evidence or conclusions must measurable with a system that is as unbiased as possible, replicable, and conforms to the Natural Laws as defined through this same process. Anecdotal evidence is any set of conclusions that may make sense to the observer, but cannot be measured, replicated, or confirmed otherwise.

They are basically equally acceptable. It's not like axioms and stuff HAVE to come from scientific evidence, they can happen by sensory observation, thinking, etc. Scientific evidence is more translatable into research format, and that is why it is preferred, but people who are reasonable know what is fallacious and implausible, and what is not.

Ok I'll give an example where the anecdotal evidence changed the science. For decades into the late 80's early 90's scientists were adamant that anabolic steroids didn't help increase strength or muscle mass (yeah I know but this WAS an accepted scientific truth) . Of course all the strength athletes, bodybuilders and other meatheads knew better: they took anabolic steroids and they got stronger and bigger. The scientists were wrong because they were performing their experiments wrong. When you administer exogenous testosterone, outside of the body's own production, the body's feedback loops sense the elevated levels of testosterone and shuts down its own production so the only circulating testosterone comes from the exogenous source. In their experiments the scientists were administrating enough testosterone to shut down the the body's own production, or endogenous testosterone, but what that left was a testosterone level the same or even lower than the original endogenous amount. So of course no increase or even a decrease in circulating testosterone will not create bigger or stronger muscles. This only changed after the scientists recognised the overwhelming anecdotal evidence and some of the meatheads themselves became scientists. Unfortunately many in the lifting community lost respect and trust in scientists because of this. Now of course this doesn't mean the scientific method is wrong but it is a example where the scientists were wrong and the anecdotal evidence was right.

I once came down with a terrible virus, headache, aching all over, extreme lethargy, everything - it was nasty. So my mum made me a cup of lemon and honey tea to try and cheer me up. A couple of hours later I felt great, the virus was gone! So it seems lemon and honey tea can cure a horrible infection right? Of course not, that's just a single anecdote, not proper science. Do the same experiment with 200 people with the same virus, get my mum to make 100 of them some tea and I bet you there's no difference between the tea drinkers vs tea abstainers!

Scientific evidence is collected by experimentation, repeated to verify and then put out for peer review to be once again verified until it becomes scientific fact.

Anecdotal evidence comes from "yea it must be true because Dave down the pub says it is and he is proper clever innit and knows about dese fings"

Just google it and you will know.

Well, this woman at the bus stop told me...