Mathematically, "either total" is 4%~6% better than "take new total".
I made a spreadsheet where I compare this using a d6. The columns represent the value you rolled on a d6, so 1 to 6. The rows represent the value rolled on the second d6 (if you did re-roll). The value in the matrix is the actual end-value taken from rolling the die/dice.
If you get to choose either total, you will always roll twice and pick the best, as shown below. The average result here is 4.47.
If you pick the new total, it only makes sense to re-roll when you roll a 1, 2, or 3 on the first die (since those values are worse than the average value for a d6). As you see below, in the columns for 4, 5, and 6, you don't re-roll, and keep the value. On the others, you re-roll and get the new value (1 to 6). The average here is 4.25.
So, on a d6, if choosing "either total" represents on average a 4.47, and taking the "new total" represents on average a 4.25, then there is a 5% improvement in being able to choose either total.
Running a simple python script shows you the improvements for:
- d4: 4.17%
- d6: 5.23%
- 2d6: 5.01%
- d8: 5.68%
- d10: 5.93%
- d12: 6.08%
- d20: 6.35%
That being said, this assumes there is no limitation on using this feature. If it is more limited, you might just re-roll when you get low values, and that will skew the averages I've shown. As R.. pointed out,
"New total" also has much higher risk, making it reasonable to decline to use it in cases where your initial roll is somewhat below the mean if a very low re-roll would be catastrophically worse than the initial low roll. Both the risk itself and any hesitance to take a re-roll due to the risk are factors that increase the advantage of the "choose either roll" variant.