3 added 2 characters in body

The Langlands group is not meant to be the motivic Galois group; rather, it is larger (in Langlands Langlands's original formulation), or alternatively not an algebraic group, but a locally compact group which has some kind of underlying algebraic avatar (this is the more recent, indeed current, formulation, due to Kottwitz), so that one can speak of both continuous and algebraic representations.

A toy model to think about is the group $\mathbb C^{\times}$, and the difference between representations of $\mathbb C^{\times}$ just as a topological group, as opposed to $\mathbb C^{\times}$ thought of as a real algebraic group (i.e. thought of as the restriction of scalars of $\mathbb G_m$ from $\mathbb C$ to $\mathbb R$).

To see this example arising in real life, one can think about the difference between arbitrary and algebraic (type $A_0$ in Weil's terminology) Hecke characters for some number field $F$. (This is the theory for $1$-dimensional reps. of the Langlands group $\mathcal L_F$.) The relevant topological group is the idele class group of $L$, while the corresponding algebraic group is what Langlands calls the Serre group in his Ein Maerchen paper (and which is studied, but with different notation and terminology, in Serre's Abelian $\ell$-adic reps. book).

A harder example can be obtained by comparing the global Weil group over a number field to the Taniyama group over this field. This is discussed in Ein Maerchen, and in the book of Deligne, Milne, Ogus, and Shih, Hodge cycles, motives, and Shimura varieties. (This is the theory obtained by combining $1$-dimensional reps. with finite image reps. of arbitrary dimension.) (See this answer for more on Weil groups.)

2 added 3 characters in body

The Langlands group is not meant to be the motivic Galois group; rather, it is larger (in Langlands original formulation), or alternatively not an algebraic group, but a locally compact group which has some kind of underlying algebraic avatar (this is the latermore recent, and indeed current, formulation, due I think to Arthur)Kottwitz), so that one can speak of both continuous and algebraic representations.

A toy model to think about is the group $\mathbb C^{\times}$, and the difference between representations of $\mathbb C^{\times}$ just as a topological group, as opposed to $\mathbb C^{\times}$ thought of as a real algebraic group (i.e. thought of as the restriction of scalars of $\mathbb G_m$ from $\mathbb C$ to $\mathbb R$).

To see this example arising in real life, one can think about the difference between arbitrary and algebraic (type $A_0$ in Weil's terminology) Hecke characters for some number field $F$. (This is the theory for $1$-dimensional reps. of the Langlands group $\mathcal L_F$.) The relevant topological group is the idele class group of $L$, while the corresponding algebraic group is what Langlands calls the Serre group in his Ein Maerchen paper (and which is studied, but with different notation and terminology, in Serre's Abelian $\ell$-adic reps. book).

A harder example can be obtained by comparing the global Weil group over a number field to the Taniyama group over this field. This is discussed in Ein Maerchen, and in the book of Deligne, Milne, Ogus, and Shih, Hodge cycles, motives, and Shimura varieties. (This is the theory obtained by combining $1$-dimensional reps. with finite image reps. of arbitrary dimension.) (See this answer for more on Weil groups.)

A toy model to think about is the group $\mathbb C^{\times}$, and the difference between representations of $\mathbb C^{\times}$ just as a topological group, as opposed to $\mathbb C^{\times}$ thought of as a real algebraic group (i.e. thought of as the restriction of scalars of $\mathbb G_m$ from $\mathbb C$ to $\mathbb R$).
To see this example arising in real life, one can think about the difference between arbitrary and algebraic (type $A_0$ in Weil's terminology) Hecke characters for some number field $F$. (This is the theory for $1$-dimensional reps. of the Langlands group $\mathcal L_F$.) The relevant topological group is the idele class group of $L$, while the corresponding algebraic group is what Langlands calls the Serre group in his Ein Maerchen paper (and which is studied, but with different notation and terminology, in Serre's Abelian $\ell$-adic reps. book).
A harder example can be obtained by comparing the global Weil group over a number field to the Taniyama group over this field. This is discussed in Ein Maerchen, and in the book of Deligne, Milne, Ogus, and Shih, Hodge cycles, motives, and Shimura varieties. (This is the theory obtained by combining $1$-dimensional reps. with finite image reps. of arbitrary dimension.) (See this answer for more on Weil groups.)