Are isomorphic structures really indistinguishable? I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different with respect to some property and I just wanted to know more about it. 
EDIT: I try to add clearer informations about what I want to talk about. In practice, when we talk about some structured set, we can view the structure in more different ways (as lots of you observed). For example, when someone speaks about $\mathbb{R}$, one could see it as an ordered field with particular lub property, others may view it with more structures added (for example as a metric space or a vector space and so on). Analogously (and surprisingly!), even if we say that $G$ is a group and $G^\ast$ is a permutation group, we are talking about different mathematical object, even if they are isomorphic as groups! In fact there are groups that are isomorphic (wrt group isomorphisms) but have different properties, for example, when seen as permutation groups. 

I'm not sure, if this is what you are referring to, but here goes...
There are questions that are easy to decide in one structure, but much more difficult in another isomorphic structure. The discrete logarithm problem comes to mind. The additive group
$G_1=\mathbf{Z}_{502}$ is generated by $5$, and to a given $x\in G_1$ finding a multiplier $n$ such that
$$
5n=(5+5+\cdots 5)=x
$$
is easy, as the generalized Euclidean algorithm will do it for us.
The multiplicative group $G_2=\mathbf{Z}_{503}^*$ is also cyclic of order $502$ and also generated by $5$. Yet, to a given $x\in G_2$ the problem of finding an exponent (now an exponent as the group is multiplicative) $n$ such that
$$
5^n=(5\cdot5\cdot5\cdots5)=x
$$
is more difficult. The difference in difficulty becomes more pronounced as the size of the groups grows.

The problem is that describing an isomorphism is not enough to translate a question from one structure to the other, if you cannot also describe its inverse.

