According to the Gramophone, the only competitive pairing of these two concertos comes from Mischa Maisky (DG). Rostropovich (Sony) remains the premier proponent of the first concerto, and Wispelwey (Channel Classics) has released another remarkable version. I would probably rate Maisky a good third choice and put Schiff in fourth, in order of my preference for these recordings.
This is not to say that Schiff is in any way an inferior cellist. His unique sound recalls the white-hot vibrato of legendary violinist Jascha Heifetz, and lends a great energy to his performance. Indeed, Schiff seems to share a few interpretative ideas with Heifetz, who also allowed music speak for itself more often than risking "over-interpretation". This approach works particularly well in the moderato, which in Schiff's hands carries more forward momentum than in any of the other mentioned recordings.
On the other hand, his cadenza is the most boring I have ever heard. Before I acquired Schiff's recording, I had considered Rostropovich's cadenza to be very straight-forward (though nevertheless excellent). But Schiff's adherence to metronomic rhythms is truly quite puzzling, especially in contrast to the vivacity of his solo Bach.
I have no such quibbles with the second concerto. In this more symphonic work, Schiff feels a bit less stiff. The larger-scale orchestral writing comes across as better balanced with the cello than in the soloist-heavy first concerto. It flows very organically through a wider range of dynamics and orchestral colors, and the soloist and conductor seem to be of one mind. Again, this recording seems charged with greater momentum than its competition from Maisky.
So, all told, we have a very fine first concerto and a definitive account of the second. The recorded sound is slightly lacking in low-end impact, but sounds just fine if you turn up the volume. Complement this with either Rostropovich or Wispelwey in the first concerto, and you will have the full measure of both concertos - the two finest works dedicated to Mstislav Rostropovich, and two of the greatest cello concertos ever composed.