> Why do these two methods give different results and which one is correct?

Both are incorrect; but the second more, I'd say. As H.Tofaili pointed out, both *often* are very similar, since every of them makes some approximation which is only always allowed in some limit. 

 - In your second approach you assume, that there is no heat conducted across the (imagined) cut between the cylinder and the hemispheres. But it will. It may be very little in most cases, but you can see it's there in an extreme case: 

  take a thick spherical shell and insert a very thin cylinder (great $R_f-R_i$, small $\ell$). It is in practice still a shell and will transport heat radially. Then with $\frac{R_i}{R_f} \rightarrow 0$ arbitrarily little of the heat starting in the cylinder will end up in it, it will all cross the border. 

 - In your first approach you assume, that the surfaces of equal temperature are all of the same form and equally spaced. This example is not as convincing as the above (I mean, here the assumption doesn't get *arbitrarily* wrong), but consinder:

  Take now a thin ($R_f\approx R_i$) figure. The central parts of the cylinder and the hemispheres are far from the other form respectively, so the heat conduction in them is decoupled from each other and behaves just as in a cylinder or hemisphere. The mean temperature will be at the geometrical mean of the radii in the cylinder and at the harmonical mean of the radii in the hemisperes. 

So every approximation has a limit in which it is bad. But I think, the first approximation is "better", it looks so "second-order", and as I said, I don't see how to make it arbitrarily wrong. But this get sheer intuition now...