Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map.

But there is a foundational paper on the topic you seek:

Brendan Hassett, Yuri Tschinkel, Geometry of equivariant compactifications of ${\bf G}_a^n$. Internat. Math. Res. Notices 1999, no. 22, 1211-1230.

There is a big difference between the two theories: For any smooth projective variety, if it has a structure as a toric variety then this structure is unique (up to a suitable equivalence). Essentially this is because any two maximal tori in a reductive algebraic group over an algebraically closed field are conjugate.

But for equivariant compactifications of $\mathbb{G}_a^n$ there can be many such structures. In fact Hassett and Tschinkel show that $\mathbb{P}^6$ has infinitely many such structures; in fact a continuum's worth, so the set of possible structures cannot be described by any discrete combinatorial data.

Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map.

But there is a foundational paper on the topic you seek:

Brendan Hassett, Yuri Tschinkel, Geometry of equivariant compactifications of ${\bf G}_a^n$. Internat. Math. Res. Notices 1999, no. 22, 1211-1230.

There is a big difference between the two theories: For any smooth projective variety, if it has a structure as a toric variety then this structure is unique (up to a suitable equivalence). Essentially this is because any two maximal tori in a reductive algebraic group over an algebraically closed field are conjugate.

But for equivariant compactifications of $\mathbb{G}_a^n$ there can be many such structures. Hassett and Tschinkel show that $\mathbb{P}^6$ has infinitely many such structures; in fact a continuum's worth, so the set of possible structures cannot be described by any discrete combinatorial data.