Hi there,
the two models are similar indeed, the only difference is that you allow for an intercept in the linear log-log model. You can derive this quite easily:
power model:
(1) ~ y = x^a
linear log-log model:
(2)~log(y) = a \cdot log(x) + b
(3)~log(y) = log(x^a) + b
(4)~y = x^a + e^b
(5)~y = x^a + c
For this reason you can compare the models more directly by using only lm() and using once a version with and without (add + 0 to you formula) intercept. In the latter case, the coefficient should match your power model. Then you can compare the models using summary(). Alternatively, allow for the intercept as well in the power model by adding a second variable to your nls() formula as per (5).
I am also no statistician so I cant really advise further than that. I hope this helps already however.
Best of luck,
Valentin