Functional programming vs Object Oriented programming
You don't necessarily have to choose between the two paradigms. You can write software with an OO architecture using many functional concepts. FP and OOP are orthogonal in nature.
Take for example C#. You could say it's mostly OOP, but there are many FP concepts and constructs. If you consider Linq, the most important constructs that permit Linq to exist are functional in nature: lambda expressions.
Another example, F#. You could say it's mostly FP, but there are many OOP concepts and constructs available. You can define classes, abstract classes, interfaces, deal with inheritance. You can even use mutability when it makes your code clearer or when it dramatically increases performance.
Many modern languages are multi-paradigm.
Recommended readings
As I'm in the same boat (OOP background, learning FP), I'd suggest you some readings I've really appreciated:
Functional Programming for Everyday .NET Development, by Jeremy Miller. A great article (although poorly formatted) showing many techniques and practical, real-world examples of FP on C#.
Real-World Functional Programming, by Tomas Petricek. A great book that deals mainly with FP concepts, trying to explain what they are, when they should be used. There are many examples in both F# and C#. Also, Petricek's blog is a great source of information.
When do you choose functional programming over object oriented?
When you anticipate a different kind of software evolution:
Object-oriented languages are good when you have a fixed set of operations on things, and as your code evolves, you primarily add new things. This can be accomplished by adding new classes which implement existing methods, and the existing classes are left alone.
Functional languages are good when you have a fixed set of things, and as your code evolves, you primarily add new operations on existing things. This can be accomplished by adding new functions which compute with existing data types, and the existing functions are left alone.
When evolution goes the wrong way, you have problems:
Adding a new operation to an object-oriented program may require editing many class definitions to add a new method.
Adding a new kind of thing to a functional program may require editing many function definitions to add a new case.
This problem has been well known for many years; in 1998, Phil Wadler dubbed it the "expression problem". Although some researchers think that the expression problem can be addressed with such language features as mixins, a widely accepted solution has yet to hit the mainstream.
What are the typical problem definitions where functional programming is a better choice?
Functional languages excel at manipulating symbolic data in tree form. A favorite example is compilers, where source and intermediate languages change seldom (mostly the same things), but compiler writers are always adding new translations and code improvements or optimizations (new operations on things). Compilation and translation more generally are "killer apps" for functional languages.