Q:

Performance advantages of using methods inside of classes verses data structures with libraries of functions?

Basically is the only advantage of object oriented languages the improved understanding of a programs purpose?
Do the compilers of object oriented languages break apart the objects into structures and function libraries?

A:

Basically, yes. The only advantage is improved understanding of code.
For some languages the OO version is the same as the non-OO version after compilation. Perl for example. For the majority of cases the OO version is much slower than the non-OO version. With very rare exceptions, non-OO languages are always faster than OO languages.
But in general, most experienced programmers will tell you not to worry about the performance differences between OO and non-OO languages (or Lispers will tell you not to worry about the performance difference between procedural and functional languages). This is because you should never, ever, ever, underestimate the importance of understanding code.
These days we rarely talk about it anymore because we've gotten used to using very high level languages - be it OO or functional or multi-paradigm or metaprogramming. But back in the 80s and 90s there was what was then known as the software crisis. What was the software crisis? It's basically the fact that most software projects were never completed!
The software crisis affected all sectors of the industry: from military radar systems, to games to commercial operating systems. The consumers called them vaporware. They were projects that were too ambitious.
But these days there are lots of very ambitious and impressive projects that manage to reach at least beta versions (and for web2.0 beta is good enough for public consumption). Part of the reason is that we now understand requirements engineering better and we also understand the process of software development better. But part of it is also because we have better tools to actually understand what we're doing. And OO is part of that toolset.

