Why is it okay to say two groups are the same, when they're really isomorphic? This is a pet peeve of mine.  Both my textbook and my instructor, often say $G$ is $G'$, when really what they mean is $G$ is isomorphic to $G'$.  Why is this an accepted convention?
 A: Because the real object is only defined up to an isomorphism. The thing you think is $G$ is really just a model, as is $G'$.
A: It is a question of language and of ontology (the essence of what a thing "is").
An Isomorphism can be seen as making a connection between two different aspects of the same abstract reality, thus illuminating its "true" nature more fully than either aspect.
Alternatively:
An Isomorphism can show us that two distinct entities with their separate realities look the same when seen from a particular perspective.
The concept of isomorphism does both jobs - hence the ambiguity - which is rarely a problem, unless you can give a specific example ... ?
A: Something I find useful is to think about abstract groups and realizations of each of them.
You could say that an abstract group is just a collection of objects with particular relations holding among them, defined through the group product (whatever it may be). You can completely specify an abstract group by writing down the multiplication table or its presentation (not an easy task, generally speaking).
Following this approach, you can say that 


*

*there is only one group of three elements: its elements are $e,a,b$, where $e\cdot a = a\cdot e = a$, $e\cdot b = b\cdot e = b$, $a\cdot a = b$, $a\cdot b=b\cdot a = e$, $b\cdot b = a$ (sorry if I didn't write nicely the multiplication table). Then its presentation is $\langle a\;|\;a^3=e\rangle$ and you call it "cyclic group".

*there are two groups of four elements: one has the presentation $\langle a\;|\;a^4=e\rangle$ (cyclic group); the other is given by $\langle (a,b)\;|\;a^2=b^2=(ab)^2=e\rangle$ (dihedral group).


Now observe that you can interpret the group of order 3, usually denoted by $\mathbb{Z}_3$, as the set $\mathbb{Z}/3\mathbb{Z}$ of residue classes with addition$\mod 3$, or the set of the complex cubic roots of unity with the ordinary product over $\mathbb{C}$.
Similarly, the cyclic group $\mathbb{Z}_4$ is the group $\mathbb{Z}/4\mathbb{Z}$ of residue classes$\mod 4$, or the group of fourth roots of unity. The dihedral group $D_2$ is usually referred to as the group of simmetries of a segment (i.e. a degenerate regular polygon with two sides).
All of these groups, then, admit different realizations in terms of matrices: these are usually called group representations (e.g. you make use of rotation and reflection matrices to construct the 2 dimensional representation of every dihedral group). The sets of matrices used to build a representation are subgroups of $\text{GL}(n)$.
Every realization of a group is isomorphic to all the others.
Hope this helps!
A: Isomorphism is just a renaming of elements.
A: It is the central insight of mathematics of structures that things that seem to be different are "really" the same thing. It doesn't help understanding if you are not able to switch the usage of "same" according to context. After all, each pair of objects is different, so what would 1+1 actually mean? And no, it is not a good idea to try to do mathematics as a formal game without any meaning.
A: Ask yourself what a group really is. It is a set together with an operation/composition that satisfies certain relations. In many ways it is the composition is what determines all the properties of the group. If you for example ask about the number of subgroups of the group or whether or not there are any normal subgroup or if the group is abelian, then it all depends on the composition. If you change the underlying set by just calling the elements something else, the all these properties remain the same. This is all the same when we for example talk about the representation theory of a group. The number of irreducible representations of a group remain the same.
However, the definition of a group is not a "model" (as in another answer). I think that it is wrong to say that a group is always defined up to isomorphic. If that was so, then we wouldn't need to talk about groups being isomorphic ...
So in the end we can think of two isomorphic groups as being the same group even though they strictly speaking are not.
Can one make this formal? Yes, one can simply talk about the set of groups and then define an equivalence relation where $G \sim H$ if they are isomorphic. Then you can "mod out" by this equivalence relation. So you get a set of equivalence classes. This would justify talking about isomorphic groups as being the same.
A: Clearly, they are not the same unless the whole system is adjusted so that they are made to be the same. Okay, what I mean by this is, for example, that when you construct $\mathbb{R}$ from $\mathbb{Q}$, you are actually constructing a new object $\mathbb{R}$ that does not at all include $\mathbb{Q}$ as a subset. But then the thing is that this new constructed object $\mathbb{R}$ contains a subset which behaves exactly in same the way as $\mathbb{Q}$ does, that is, isomorphic. So in this case, we wholly discard the previous $\mathbb{Q}$ but then adopting the whole new system $\mathbb{R}$ we redefine that subset as $\mathbb{Q}$ (using the same symbol). Note that though the previous rational number system and the new rational system are denoted by the same symbol, they are never the same logically.
Okay back to actual point. The reason why we simply say 'is' or consider them as 'equal' is only for convenience, the convenience to avoid all the detailed formalisms. If you need to deal with those groups in the fully formalized rigorous context, you should not make them as the same. But often times, you can prove just 'intuitively', and since to make them into a fully formalized version is a mere labor (though it maybe very long time-consumming labor), you just avoid them.
A: It depends on context. Sometimes it's ok to say they are the same and sometimes it isn't.
An isomorphism is just a renaming of elements, so as long as the names don't actually mean anything or are unconnected to anything then it's ok. Structural information about a group such as how many subgroups it has doesn't depend on the names of the elements.
But as soon as the names of the elements are related to something else then the labelling of elements does matter. See for example the question about whether two rings can be non-isomorphic to each other even though the operations from one ring are isomorphic to the operations in the other ring. 
Also the question "How many automorphisms does a group have?" requires you to both distinguish between relabellings and equate them (as isomorphic) at the same time.
