Why do both the abstract class and interface exist in C#?

It's not a trivial question, it's a very good question and one I always ask any candidates I interview.
In a nutshell - an abstract base class defines a type hierarchy whereas an interface defines a contract.

You can see it as is a vs implements a.
i.e Account could be an abstract base account because you could have a CheckingAccount, a SavingsAccount, etc all which derive from the abstract base class Account. Abstract base classes may also contain non abstract methods, properties and fields just like any normal class. However interfaces only contain abstract methods and properties that must be implemented.

c# let's you derive from one base class only - single inheritance just like java. However you can implement as many interfaces as you like - this is because an interface is just a contract which your class promises to implement.

So if I had a class SourceFile then my class could choose to implement ISourceControl which says 'I faithfully promise to implement the methods and properties that ISourceControl requires'

This is a big area and probably worthy of a better post than the one I've given however I'm short on time but I hope that helps!

Well, an abstract class can specify some implemetation, but usually not all of it. (Having said which, it's perfectly possible to provide an abstract class with no abstract members, but plenty of virtual ones which with "no-op" implementations). An interface provides no implementation, merely a contract.

You could certainly argue that if multiple inheritance of classes were permitted, interfaces would be largely pointless.

Personally I don't get hung up on the whole "is-a" vs "can-do" distinction for inheritance. It never gives me as good an intuition about what to do as just playing around with different ideas and seeing which ones feel the most flexible. (Then again, I'm very much a "favour composition over inheritance" guy...)

EDIT: Just as the most convenient way of rebutting lbushkin's third point in his comment... you can override an abstract method with a non-virtual one (in terms of not being able to override it further) by sealing it:

public abstract class AbstractBase
{
    public abstract void Foo();
}

public class Derived : AbstractBase
{
    public sealed override void Foo() {}
}

Classes deriving from Derived cannot override Foo any further.

I'm not in any way suggesting I want multiple inheritance of implementation - but if we did have it (along with its complexity) then an abstract class which just contained abstract methods would accomplish almost everything that an interface does. (There's the matter of explicit interface implementation, but that's all I can think of at the moment.)