When should one use Lagrange Multipliers as opposed to calculating the ratio of Marginal Utilities for MRS?
Lagrange multipliers not only incorporate constraints on a maximization/minimization problem, but the multiplier itself can have a meaningful interpretation. For example, say we have the problem:

$$\max_{x, m}: U(x, m) = m + \phi(x) \quad s.t. m + px = y$$

Where $m$ is a numeiare good, $p$ is the price of good $x$, and $y$ is income.

Now you could just rewrite the constraint as

$$m = y - px$$

and substitute that into the original maximization problem. But you can't always do this easily. So you can also solve with a Lagrangian.

$$\mathcal{L} = m + \phi{x} - \lambda(m + px - y)$$

Solving for $\lambda$ will give you the shadow price of income, or the marginal price of income. That is, it represents the rate of increase in maximized utility as income is increased. You also need to find $\lambda$ to get a nice full Marshallian demand, or to relate indirect utility to Marshallian demand in Roy's identity.

Even for problems that aren't utility maximization, $\lambda$ is still representing the rate of change of the quantity being optimized as a function of the constraint variable.

Here's a useful example with the firm maximization problem and bribing problem.