Let $H$ and $G$ be Lie groups, one has, by definition, $H$ is a Lie subgroup of $G$ if exists a smooth injective homomorphism $f:H\longrightarrow G$.
This is automatically an immersion, but it is not an embedding; it is, instead, an embedding if $H$ is closed.
If i consider the case in which $f$ is an inclusion map, i have that the Lie algebra of $H$ is a Lie sub-algebra of the Lie algebra of $G$ and it is characterized by the exponential map of $G$ (this due to naturality of the exponential map).
What i gain if $H$ is closed?
The only thing i found is that there is only one differentiable structure that makes $H$ a Lie group and that this is the one that comes from $G$ (a sort of universal property).
Can someone please help me with examples to understand why this is so important?
Is there some other property i have missed that can help me to go inside the conceptual difference?