The approximation of the matrix exponential times a vector, \(\omega(\tau)=\exp(-\tau A)v\), where \(A\) is a large and sparse symmetric positive semidefinite matrix and \(\tau\) is a positive constant, is required in many discretization schemes of differential equations (ordinary and partial). Two well-known methods for dealing with this problem are the Lanczos method and the shift-and-invert Lanczos method. Both involve Krylov subspaces and generate a sequence of vectors \(\omega_m(\tau)\) converging to \(\omega(\tau)\). The most used bounds for the absolute error \(\|\omega_m(\tau)-\omega(\tau)\|\) depend mainly on the norm of \(\tau A\). The main goal of this paper is to derive new error bounds showing, in particular, that the condition number of \(A\) has also a key role in the convergence of the methods. Some practical examples are analyzed in detail to illustrate the theoretical results.