Historical weather is able to predict future weather MUCH better than historical financial data can predict future financial data. Technologies of trading/investing change quickly and market mechanics of 80's is very different from current behaviour. Weather has periodicity and pretty smooth predictable patterns, unlike financial series where you can observe spikes, lack of mean reversion etc.
Good quality weather observations can go back to early 1900s cmp. to financial data which usually spans 2 decades or even less (again, early data wouldn't make any sense anyway). So it has much more training data.
Certainly, weather prediction takes into account not only the time series values (same for financial predictions). But even the most naive approach "predict December weather as average of past 10 years Decembers" will give pretty good approximation (financial prediction like this will be complete nonsense). There are laws of physics which come into play and they are much more strict than laws of financial markets - after all, the amount of randomness in cyclones and winds is much less than in market movements.
To improve your model you need to firstly understand the mechanics of time series you are trying to predict and base your model on it, not vice versa. Most likely, taking into account only the stock prices you won't be able to predict anything - try to see other parameters and indicators, maybe looking at volumes and open interest, trying to find some correlations there. It is much harder than it seems because if it would be easy everyone would do it which would force prices to converge to 'fair values', thus making it useless again. I suggest not making your model more complicated, but rather spending your time researching market mechanics and only then going into building it - you won't get far by just feeding data into some standard predictor and expecting it to make good predictions.
