When to use a lagrange multiplier?
I wish I could leave this as a comment, but the first formulation is a solution to a maximisation problem of the form
$$
\max_{C_t} \sum_{t} \beta^{t} u(C_t) \\
\text{s.t. } \quad C_t +K_{t+1} \leq (1-\delta)K_{t} + w_{t} + r_tK_t \quad \forall t 
$$

The second formulation is a solution to a maximisation problem of the form
$$
\max_{C_t} \sum_{t} \beta^{t} u(C_t) \\
\text{s.t. } \quad \sum_{t} C_t \leq \sum_{t} (A_0 +Y_t) 
$$

In a way, the first is a sequential problem, while the second is a time zero problem. In some instances the approaches are equivalent, but I'm not an expert on this.