comparison between two monadic definitions for an operad According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is equivalent to a monad in the bicategory of modified spans, diagrams of the form $T1\leftarrow C\to 1,$ where $T$ is the free monoid functor.
Is there some connection between these two monads, other than the fact that they unravel in base cases to the definition of an operad?
 A: Well the two monads are quite different: in May definition you deal with an actual monad in $\mathbf {Cat}$ (i.e. a strict-$2$-category) while in the second case you work with monads in the bicategory of sets and spans in $\mathbf {Set}$ (actually this second kind of monad is a $T$-operad).
Nonetheless the two monads are actually linked together: May's monad is the monad functor part used to build the algebras of Leinster's $T$-operad.
Here follows the details of the construction.
Working a little bit with Leinster definition of operads you can see that the span diagram $T1 \leftarrow C \rightarrow 1$ is characterized completely by the left arrow $p \colon C \to T1\cong \mathbb N$ and such arrow can be seen as the family of elements 
$$C_{i}=\{ c \in C \mid p(c)=i \in T1\}$$ 
(that's basically an application of the equivalence between fibrations in $\mathbf{Set}$ and indexed families of sets).
Then the functor $\mathcal C \colon \mathbf {Set} \to \mathbf {Set}$ that gives May's monad can be recoverd as 
$$\mathcal C(X) = \bigsqcup_{i \in T1} C_i \times X^i$$
If your reference for $T$-operads is Leinster's Higher Operads Higher Categories then you should recognize that this is the monad induced by a $T$-operad whose algebras are the algebras of the $T$-operads.
