Are there situations when regarding isomorphic objects as identical leads to mistakes?

Inside of the complex numbers there are lots of examples of distinct fields which are isomorphic. For instance, there are three subfields of the form ${\mathbf Q}(\alpha)$ where $\alpha^3 = 2$: take for $\alpha$ any of the three complex cube roots of 2 and you get a different subfield. What are the consequences of treating them as literally equal? You can't make any sense of Galois theory if you do that! Similarly, all $p$-Sylow subgroups of a finite group are isomorphic (since conjugate subgroups are isomorphic groups), but it would kind of destroy a lot of the content of the Sylow theorems by trying to say the $p$-Sylow subgroups are identical.

More generally, anytime you have isomorphic but unequal objects inside a larger object, it can lead to confusion if not outright incomprehensibility if you try to regard them all as identical. (There was a paper by Chevalley about unit groups in number fields where he made a genuine error by an abuse of the "square root" notation and I think one might be able to express the mistake in the form of an isomorphism being confused with an equality, but I'd have to look at the paper again to be sure about this.)

The word isomorphism does not emphasize that two objects are different; any group or vector space admits an isomorphism with itself using the identity map. The word emphasizes that in a structural way the two objects look like each other even though they are not literally the same. Never say two objects are identical if they are not actually identical. Having said that, I must admit that in mathematics one meets phrases like "since $X$ and $Y$ are isomorphic we can identify $X$ with $Y$" and then $X$ is replaced with $Y$. The usefulness of doing this depends on the application you have in mind. Note, however, that replacing $X$ with $Y$ is not saying that $X$ and $Y$ are the same thing.

This question sounds like it is being asked by someone who hasn't had a lot of experience with isomorphisms and is trying to get a feel for what it means. In a year or two, after seeing more appearances of the concept and its uses, you'll get a better feel for it, but for now do not think the word isomorphism is a synonym for identical.

A slightly more general answer: if the objects in question have no non-trivial automorphisms (i.e. non-identity isomorphisms from itself to itself) then no danger will come from treating isomorphisms as the identity. (E.g. the real numbers has no automorphisms as a field, and so any two copies of the real numbers can be unambiguously identified as fields.)

But if $X$ and $Y$ are isomorphic but Aut($Y$) (or, equivalently, Aut($X$)) is non-trivial, then the identification of $X$ and $Y$ is not uniquely determined (because you could always postcompose with a non-trivial automorphism of $Y$, or precompose with a non-trivial automorphism of $X$, to obtain a different identification), and hence in this case there is not an unambiguous identification. (A typical example of this is the isomorphism between a finite-dim'l vector space and its dual; there is not a uniquely determined such isomorphism, and so one should not identify the two, although they are isomorphic.)

In some contexts, although there is not a uniquely determined isomorphism, there is a canonical choice, e.g. the identification of a finite dimensional space with its double dual. Such isomorphisms are normally described in the language of natural isomorphisms between functors (e.g. the identify functor and the double duality functor are naturally isomorphic as functors from the category of finite-dimensional vector spaces --- over some given field --- to itself.) Often it is safe to identify objects that are identified by a natural isomorphism in some category theoretic framework that is suitable for the problem at hand. But even in these cases, sometimes one has to actually know an explicit description of the natural isomorphism (e.g. because you might need to make a computation involving both objects, which will of course require you to know how they have been identified). In particular, when identifying two objects via some natural isomorphism, it is good form to explicitly describe the natural isomorphism at some point, unless the natural isomorphism in question is utterly conventional (e.g. the double duality isomorphism for a finite-dimensional vector space; and even then, one might write something like "we identify $V$ and $V^{\vee\vee}$ via the usual double duality isomorphism").

Added: This answer is very similar to that of Charles Staats, which was posted while I was writing.

If $p$ is a prime, then the multiplicative group modulo $p$ is isomorphic to the additive group modulo $p-1$, which is to say it's a cyclic group of order $p-1$. It's trivial to find a generator of the additive group, but a fair bit of work to find one for the multiplicative. Related to this, it's easy to solve $ax=c$ in the additive group, using the Euclidean algorithm, but the difficulty of solving the corresponding equation $a^x=c$ in the multiplicative group - the discrete logarithm problem - is the basis for some cryptography systems.