R2, P-value, or accuracy of model? Which is more important when selecting a model? The R2 or the accuracy of the model? 
Here are 2 different models using the same data with different transformations for the Xs, the Ys are a percentage and the same in both. 
This is a classification problem in which correct classification has a higher growth rate associated with it. Therefore, what matters in reality is accuracy and growth. So for what matters in reality I'd choose B, but how do its not just better due to randomness?
Model A
Residual standard error: 0.03033 on 338 degrees of freedom
Multiple R-squared: 0.03352,    Adjusted R-squared: 0.02205 
F-statistic: 2.922 on 4 and 337 DF,  p-value: 0.02127 
AIC: -1410.358

Accuracy:
PointEst  Lower     Upper
0.6012085 0.5305505 0.6678887

Growth: 6.116476

Model B
Residual standard error: 0.03035 on 338 degrees of freedom
Multiple R-squared: 0.03032,    Adjusted R-squared: 0.01884 
F-statistic: 2.642 on 4 and 338 DF,  p-value: 0.03368 
AIC: -1413.970

Accuracy:
PointEst  Lower     Upper
0.6174699 0.5471007 0.6832359

Growth: 8.032874

As you can see, both models have the same number variables. The only difference is how I transform/treat the input variable data before running the regression. From the time where I noticed the difference between the two models, I ran them with an additional 20 observations and both performed equally as well in classification. 
 A: In general it is not fully meaningful to compare two $R^2$ when you transform $Y$ differently.  If you are only transforming $X$s it can be useful to compare $R^2$.
If you are concerned with how the $X$'s should be transformed, a more principled way to go about this is to allow the $X$s not known to be linear to be represented by smooth nonlinear functions (e.g., restricted cubic regression splines, i.e., natural splines), and not use $R^2$ to guide this process.
It is also useful to compute an accuracy measure that was not optimized during model fitting, to give another opinion about the overall fit, e.g., mean absolute error.
A: What is more important is neither R Square nor accuracy as you measure it, but that your model is structured correctly.  As Frank pointed out when you transform variables, models are not comparable.  
One thing to watch for is the nature of your dependent variable Y.  When you have a dependent variable that consists of time series that grow over time (many do such as macroeconomic variables such as GDP, CPI, etc...); a model using a dependent variable that is a nominal level (GDP size) will always have very high R Square and low Standard Error.  However, such models are heteroskedastic by definition.  This means you can't measure accurately the statistical significance of any of your variables because the variance around any of their regression coefficients is not stable enough.  As a result the estimated Confidence Interval is wrong.  Also, such model's residual would not be randomly distributed.  They would trend.  You could measure that by looking at the Durbin Watson score or simply by measuring the autocorrelation of the residuals (because such residuals are strongly autocorrelated). 
In the above case, you would have to change your dependent variable from GDP level to % change in GDP from one period to the next.  This eliminates heteroskedasticity, autocorrelation of residuals.  But, such a model will typically have a far lower R Square, higher (relative) standard error, and overall accuracy as you would measure it than the first model.  Yet, this second model is much better because it has a more robust structure and does not break the underlying assumptions of linear regression.  Meanwhile, the first model using nominal GDP would appear to be more accurate, have a better fit, etc...  Yet, it be all wrong because it does break the underlying assumptions of linear regression (the problems mentioned above). 
Here is another thought assumming both your models have a robust and comparable structure: test them using Hold Out sample.  You could rerun both your models using let's say 10 data points as Hold Out... and doing this several times with different Hold Out samples.  Then, look at which model performs best.  Performance in Hold Out sample is probably a lot more important than any statistics of fit or accuracy over the learning sample.  Given the very low R Square of both of your models, I suspect they will both perform equally badly in Hold Out.  But, in any case doing Hold Out is a good way to differentiate between models.      
