The answer depends on many details, but at a minimum you need to account for the area over which the DEMs are compared before you can draw a meaningful conclusion. I provide an overview of what to watch out for and some examples to show the analysis is simple to do when you make some conservative simplifying assumptions.
To get a correct answer, you need to understand what the "RMSE" really means. In most circumstances with DEMs, it is estimated by comparing the DEM's value at a relatively small number of widely scattered points to the "ground truth" at those points. Intuitively, it measures the typical size of deviation between what the DEM says and what the true value is.
You need to understand several things about this.
First, errors are spatially correlated. DEMs tend to present relatively smooth representations of relatively smooth elevation models--they rarely look speckled the way a low-light image does, for instance. This means that if a DEM at one location has a value, say, 0.10 meters high, then most likely its immediate neighbors have similarly high values. Typically, you have to move tens, hundreds, or even thousands of cells away until the error is no longer predictable.
Second, errors are not homogeneous. (The technical term is that they are heteroscedastic.) This means that the typical amount of error can vary according to other factors, especially slope. This is intuitive: where slopes are high, even a slight error in the position of an elevation reading translates into a relatively large error in the elevation itself, whereas where the elevation is horizontal, an error in position contributes no error to the elevation. (I analyzed this in some detail in a post at https://gis.stackexchange.com/a/56573/664.)
Third, errors are not necessarily independent. For instance, if one of the two DEMs being compared was derived from the other--or if both were derived from a common original source--then where one is too high the other is likely to be too high and where one is too low the other is likely to be too low. This is a beautiful thing when comparing DEMs point-by-point, because such errors cancel. But it's also conceivable that there will be no cancellation at all, such as when the DEMs are developed independently using completely different methods.
How, then, are we to exploit information about typical errors in DEMs? Unfortunately, it depends. In this case, with the limited information we have, you should proceed conservatively: make sure you don't underestimate the error. To this end, we must suppose that (a) the regions of the DEMs you have compared exhibit strong spatial autocorrelation; (b) the region of interest is rugged terrain; and (c) the DEMs were independently generated. In such circumstances we have to suppose that all the values in the first DEM could be uniformly off by an amount even greater than suggested by the 0.03 m RMSE; and that all the values in the second DEM could be uniformly and independently off by an amount even greater than suggested by the 0.10 m RMSE.
Under these assumptions, elementary probability theory states that the RMSE of the difference is at least the root sum of squares:
RMSE(difference) >= sqrt(0.03^2 + 0.10^2) = 0.1044.
As you can see, the error is dominated by that of the second DEM. This suggests we not worry too much about the details and hope that the value 0.1044 for the RMSE of difference is reasonable. But this error has to be applied to every cell.
As an example, consider a difference of "150.000" cubic meters. In some countries this means 1.5 E2 (one hundred fifty) cubic meters while in others it means 1.5 E5 (one hundred fifty thousand) cubic meters. I'll adopt the first interpretation -- it doesn't matter for this example. The issue is, how much area does that cover?
Suppose, for instance, the DEMs are compared over an area of 10,000 square meters. Thus, the difference of 150,000 cubic meters is an average difference of
150,000 cubic meters / 10,000 square meters = 15 meters.
To this we must apply the RMSE of 0.1044 meters to give a standard error of that difference: it is 15 plus or minus 0.1044. Evidently, in this case, the RMSE shows the difference has been estimated with a high relative precision.
The corresponding RMSE of the volume is
10,000 square meters * 0.1044 meters = 1044 cubic meters,
giving a value of 150,000 plus or minus 1044 cubic meters.
On the other hand, suppose the DEMs really were compared over an area of 1,000,000 square meters. Their average difference in this case would be
150,000 cubic meters / 1,000,000 square meters = 0.15 meters.
Now the standard error of 0.1044 is appreciable. Indeed, using the 68-95-99.7 rule, we ought to conclude that there isn't significant evidence of any real change!
The corresponding RMSE of the volume is
1,000,000 square meters * 0.1044 meters = 104,400 cubic meters,
giving a value of 150,000 plus or minus 104,400 cubic meters.
Use the conservative estimates described here to obtain an approximate sense of the net error in difference. If that error is small enough for your purposes, you don't have to sweat the details. If that error is large, then look into the possibilities of low spatial correlation and/or positive correlation between the DEMs for opportunities to reduce the error estimate. You will likely have to research the metadata of both DEMs carefully to find the information you need.