img source
I assume that linear mode analysis is meant to be linear stability analysis around equilibrium states.
It is not meant to be more general than linear mode analysis (in fact, it is supplementary between the two), it is more like a quantitative indication of your system if your system happens to have chaotic flow in the state space and, in my opinion, one of more simple method (although, there is computational cost that must be paid) of showing quantitative indication (at least numerically) that system really is have a chaotic flow in state space (by showing there is, at least, one positive Lyapunov Exponent)..
So, why Lyapunov exponent?
Linear mode analysis really restrict us to consider a flow in a really tiny state space. This is nice if your system only have equilibrium states in the form of point (and if you only consider how this equilibrium states bifurcates). The problem happens when your equilibrium states is in the form of periodic solution or quasiperiodic solution. When this happens, we could only see its qualitative behavior around point where solution passing by. This, in most case, can only be done using numerical calculation in the first place (taking Poincare Section, etc.,). Lyapunov exponent, kinda, bypass this by seeing that if your Lyapunov exponent all negative, mean it is stable equilibrium states in the form of points, and if there is one zero and all the other are negative, meant it is a periodic solution. It is more simple to interpret Lyapunov exponent.
Also, Lyapunov exponent can be calculated only by solving the system and its variational equation counterpart. In the Poincare section, although it give you pretty images, you need to choose where to slice your solution, which would make a headache even if you consider for 5, 6 state space dimension.
So, that's my take. It is complimentary between the two. You could check more of this in S. Wiggins book, Introduction to Applied Nonlinear Dynamical Systems and Chaos.