My own experience is that Lawvere theories help one "think outside the box" in ways that I really don't think are too likely with classical universal algebra. Qiaochu has already pointed to what is the key idea: that they enable one to consider models other than in $Set$. Actually, you could put it more strongly. Namely, in Lawvere's formulation, the theory is a model in this extended sense: it is the universal model. For example, the Lawvere theory of groups is the category with finite products where all you know about it is that it has a group object, and nothing more. You could take this philosophy further still: any category with finite products $C$ could be considered an algebraic theory in its own right, where a model of $C$ in a category with finite products $D$ is a product-preserving functor $C \to D$. And this conceptual expansion pays off.
Let me give a concrete example of this: consider the concept of Boolean algebra. Here the Lawvere theory is the category of finite sets of cardinality $2^n$, denoted $Fin_{2^\bullet}$. This is already a neat way to define a Boolean algebra: as a product-preserving functor
$$Fin_{2^\bullet} \to Set.$$
But it's even nicer when you complete $Fin_{2^\bullet}$ to its Karoubi envelope. It's not hard to see that if $C$ is a category with finite products, then the Karoubi envelope $\bar{C}$ also has finite products, and the models of $C$ are equivalent to models of $\bar{C}$. In the present case, the Karoubi envelope of $Fin_{2^\bullet}$ is $Fin_+$, the category of finite nonempty sets. Thus a Boolean algebra is equivalently a product-preserving functor
$$Fin_+ \to Set$$
This not only looks prettier, but it frees us from tendencies of thinking of Boolean algebras as having intrinsically to do with things like power sets and powers of 2: there's no power of 2 sticking out in this description. I call this an unbiased Boolean algebra -- we have removed the bias toward 2.
It also allows one to see that one could be biased in different ways, and see Boolean algebras, if we want, as having to do with powers of 3 instead. In other words, the Lawvere theory $Fin_{3^\bullet}$ has, by the same Karoubi envelope reasoning, the same models as $Fin_+$, so we could just as well think of a Boolean algebra as a product-preserving functor
$$Fin_{3^\bullet} \to Set$$
which gives a totally different way of seeing Boolean algebras as monadic over $Set$ -- here the "underlying sets" of such algebras have a tendency to look like structures on sets of cardinality $3^n$. If you are wondering what good that is for, I might recommend looking at the nLab article on Boolean algebras, and on the multiplicity of different ways of understanding the ultrafilter monad which this suggests. I personally have found this quite eye-opening.
Historically, the heyday of Lawvere theories was in the sixties, when the close connections with monads were ironed out. Lawvere theories are essentially the same things as finitary monads; one advantage of Lawvere theories is that it enabled one to more clearly see this connection, which in turn leads to a clear view of the story about infinitary theories and their equivalence with monads on $Set$ (at least for locally small theories). Nowadays these things are under good control, and they allow us to define things like "compact Hausdorff objects" as models of the infinitary Lawvere theory attached to the ultrafilter monad -- these too are useful. I don't think any of this would have been at all easy to see from the point of view of classical universal algebra.
The connection between theories and monads is worked out in this nLab article. Cf. also this MO answer, which goes into some more detail on the discussion about Boolean algebras.