C# offers three different ways to denote an anonymous function: Anonymous delegates, lambda expressions, and statement-bodied lambdas. Each of them are true closures and can be used to construct a "method" of a given delegate type (i.e. any one of them can be used to as a value for a given variable — interestingly, in any case, an anonymous method must be assigned to a variable (or parameter) before you can use it, unlike, say, the anonymous functions of Lisp, JavaScript, F#, or even the criminally underrated VB).
But they have some important difference:
1) Anonymous delegates (introduced in C# 2.0) allow you to elide the formal parameters when defining a method of a given type if the body of the method doesn't use those parameters. Even though this construct is basically obsolete, I still use it just for this feature.
2) Lambda expressions can be automatically transformed into expression trees, which means that you can use them to construct (awesomely powerful) fexprs[1]. But, they can't contain statements, which means you can't use them for any kind of imperative code. These are the most equivalent to Python's lambda as far as that limitation is concerned.
3) Statement-bodied Lambdas can contain statements (as the name implies), but the language won't convert them into expression trees for (you can do it manually – tedious but not at all hard). At the time that the lambda expression feature was introduced (as a member of the constellation of features that made up LINQ), the framework didn't even contain expression classes that corresponded to most of the imperative constructs, but that changed with the advent of C# 4.0 which implemented a meta-object protocol[2].
Another implementation detail (last time I checked, which has been a while) is that anonymous delegates get turned into instances of System.Multicast delegate, whereas a lambda expression either gets turned into a class with a method that corresponds to the lambda expression (and fields that correspond to closed over variables), or into an expression tree, depending on the class of the variable to which it's being assigned.
But they have some important difference:
1) Anonymous delegates (introduced in C# 2.0) allow you to elide the formal parameters when defining a method of a given type if the body of the method doesn't use those parameters. Even though this construct is basically obsolete, I still use it just for this feature.
2) Lambda expressions can be automatically transformed into expression trees, which means that you can use them to construct (awesomely powerful) fexprs[1]. But, they can't contain statements, which means you can't use them for any kind of imperative code. These are the most equivalent to Python's lambda as far as that limitation is concerned.
3) Statement-bodied Lambdas can contain statements (as the name implies), but the language won't convert them into expression trees for (you can do it manually – tedious but not at all hard). At the time that the lambda expression feature was introduced (as a member of the constellation of features that made up LINQ), the framework didn't even contain expression classes that corresponded to most of the imperative constructs, but that changed with the advent of C# 4.0 which implemented a meta-object protocol[2].
Another implementation detail (last time I checked, which has been a while) is that anonymous delegates get turned into instances of System.Multicast delegate, whereas a lambda expression either gets turned into a class with a method that corresponds to the lambda expression (and fields that correspond to closed over variables), or into an expression tree, depending on the class of the variable to which it's being assigned.
[1] https://en.wikipedia.org/wiki/Fexpr
[2] https://en.wikipedia.org/wiki/Metaobject