  Motivated by the opposite trend of maximum $T_c$ in multilayer nickelate and
cuprate superconductors, we propose an interlayer pairing scenario for the
superconductivity of the trilayer nickelate La$_4$Ni$_3$O$_{10}$ discovered
recently under high pressure. Our theory reveals intrinsic frustration of the
spin-singlet pairing formed between two outer layers and the same inner layer,
which causes strong superconducting fluctuations between layers and thus
explains the reduction of its maximum $T_c$ compared to that of the bilayer
La$_3$Ni$_2$O$_{7}$. This supports a fundamental distinction between multilayer
nickelate and cuprate superconductors associated with their potentially
different (interlayer vs intralayer) pairing mechanisms. Our theory predicts
extended $s^\pm$-wave gap structures in La$_4$Ni$_3$O$_{10}$, with varying
signs and possible nodes on different Fermi pockets. We also find an intrinsic
Josephson coupling with potentially interesting consequences to be examined in
future experiments.
