Many folks, when hearing forecasts have improved, will disagree. They can think of many times when the forecast was wrong, perhaps hilariously or disruptively so.
This is textbook availability bias. You overestimate the failure rate because the failures were memorable, especially the major failures. But the far more frequent accurate predictions are not recalled, because they don't leave as much of an emotional impression. For the same reason folks regularly overestimate crime rates, deaths from terrorism or rare diseases.
Meteorology has an entire subfield devoted to studying the quality of forecasts[0]. Before you bandy about numbers and anecdotes, stop and tell me what measure you are using. Is it percentage of degrees centigrade? Well you can't, degrees centigrade is an interval measure, not a ratio measure. If you use Kelvins, which has a mathematically meaningful zero, the percentage accuracy is suddenly very good. How do you count near misses of very intense weather? Direct hits by systems that were less intense than expected? How do you account for systems with very gradual gradients over very wide areas? How do you account for being early by an hour or late by an hour, but nailing the storm surge? How do you score for confidence? What's your rule for weighting false alarms? How important is mean error vs absolute error vs variance, and why?
Solve these and dozens of other problems of describing "accuracy" and weather forecasters might take you seriously. But until then it's probably worth accepting that they are the most effective profession of their kind and that we have a lot we can learn from them.
[0] http://www.cawcr.gov.au/projects/verification/