Your question seems to reflect the mistaken understanding that the statistical "significance" of the p-value somehow means "meaningful", "important", or "relevant to real life". This is a false but very widely held misunderstanding.
P-values are a standardized representation of how reliable the effect size measures are. I say that the p-value is "standardized" in the sense that no matter what the test statistic, whether a t score, an F, a $\chi^2$, or whatever (in the case of linear regression, it is the t score), the p-value represents it on the same scale of a number from 0 to 1 such that values closer to 0 give greater confidence that the effect size measures are reliable, that is, are not mere haphazard statistical flukes. (Of course, by convention, 0.05 is the most common cutoff measure for an "acceptable" p-value, though that should not be taken to be a magic number. The "right" cutoff for a p-value actually depends on both the sample size and the strength of the expected effect size. So, in some cases, 0.05 might not be strict enough, and in other cases, it might be too stringent.)
Effect size measures are the most important results that actually reflect what is "meaningful", "important", or "relevant to real life". In the case of linear regression, these are the coefficients and standardized coefficients (or the adjusted $R^2$ for the overall regression model).
So, with that understanding, to directly answer your question:
- The p-values reflect how much confidence you should have in the reliability of the standardized coefficients. If you take the 0.05 traditional cutoff, the only thing that the p-values tell you in your results is that all your standardized coefficients are reliable. They absolutely do not tell you how important any specific variable might be, and they should never be misinterpreted in that way. So, it is irrelevant which p-values are larger than others, as long as they are all well below 0.05.
- The standardized coefficients are what you should focus on in trying to determine which variables are more important. In regression, what they mean is that one standard deviation increase in the given variable will give the specified number of standard deviations of change in the target variable. So, in your case, one standard deviation increase in
x2 (measured in
x2's units) will increase
y by 0.24 standardard deviations (measured in
y's units); one standard deviation increase in
x3 (measured in
x3's units) will increase
y by 0.27 standardard deviations (measured in
y's units). The relative p-values of
x3 are irrelevant; they do not reflect importance; they only indicate that the estimates of
x3 both have high confidence.
That said, considering that 0.24 and 0.27 as $\beta$ values are very close, you should certainly not trust that your results mean that
x3 is conclusively more important than
x2. The numbers are too close; they should be considered approximately equal; any apparent differences here might indeed be a statistical fluke. If you really want to test which one is more important, then there are specific tests for that.