Update: You have fixed your title, so ignore the rant.
The title of your question has nothing to do with your code samples. Your question implies that one syntax is IEnumerable and the other is IQueryable, but this is incorrect. In your samples, if db.Surveys is an IQueryable, then both your samples are using IQueryable. I will try to answer both questions.
Your two code samples are just different ways of writing the same LINQ queries (assuming they are well-written). The code in sample 1 is just shorthand for the code in sample 2. The compiler treats the code in both samples the same way. Think of the way the C# compiler will treat int? the same as Nullable<System.Int32>. Both the C# and VB.Net languages provide this shorthand query syntax. Other languages might not have this syntax and you would have to use the sample 2 syntax. In fact, other languages might not even support extension methods or lambda expressions, and you would have to use an uglier syntax yet.
Update:
To take Sander's example further, when you write this (query comprehension syntax):
var surveyNames = from s in db.Surveys select s.Name
You think the compiler turns that shorthand into this (extension methods and lambda expression):
IQueryable<string> surveryNames = db.Surveys.Select(s => s.Name);
But actually extension methods and lambda expressions are shorthand themselves. The compilers emits something like this (not exactly, but just to give an idea):
Expression<Func<Survey, string>> selector = delegate(Survey s) { return s.Name; };
IQueryable<string> surveryNames = Queryable.Select(db.Surveys, selector);
Note that Select() is just a static method in the Queryable class. If your .NET language did not support query syntax, lambdas, or extension methods, that is kinda how you would have to write the code yourself.
What are the benefits of using one style over the other?
For small queries, extension methods can be more compact:
var items = source.Where(s => s > 5);
Also, the extension method syntax can be more flexible, such as conditional where clauses:
var items = source.Where(s => s > 5);
if(smallerThanThen)
items = items.Where(s => s < 10);
if(even)
items = items.Where(s => (s % 2) == 0);
return items.OrderBy(s => s);
In addition, several methods are only available through extension method syntax (Count(), Aggregate(), Take(), Skip(), ToList(), ToArray(), etc), so if I'll use one of these, I'll usually write the whole query in this syntax to avoid mixing both syntaxes.
var floridaCount = source.Count(s => s.State == "FL");
var items = source
.Where(s => s > 5)
.Skip(5)
.Take(3)
.ToList();
On the other hand, when a query gets bigger and more complex, query comprehension syntax can be clearer, especially once you start complicating with a few let, group, join, etc.
In the end I will usually use whichever works better for each specific query.
Update: you fixed your title, so ignore the rest...
Now, about your title: With respect to LINQ, IEnumerable and IQueryable are very similar. They both have pretty much the same extension methods (Select, Where, Count, etc), with the main (only?) difference being that IEnumerable takes Func<TIn,TOut> as paremeters and IQueryable takes Expression<Func<TIn,TOut>> as parameters. You express both the same way (usually lamba expressions), but internally they are completely different.
IEnumerable is the doorway to LINQ to Objects. The LINQ to Objects extension methods can be called on any IEnumerable (arrays, lists, anything you can iterate with foreach) and the Func<TIn,TOut> is converted to IL at compile time and runs like a normal method code at run time. Note that some other LINQ providers use IEnumerable and so are actually using LINQ to Objects behind the scenes (LINQ to XML, LINQ to DataSet).
IQueryable is used by LINQ to SQL, LINQ to Entities, and other LINQ providers which need to examine your query and translate it instead of executing your code directly. IQueryable queries and their Expression<Func<TIn,TOut>>s are not compiled into IL at compile time. Instead an expression tree is created and can be examined at run time. This allows the statements to be translated into other query languages (for example T-SQL). An expression tree can be compiled into a Func<TIn,TOut> at run time and executed if desired.
An example that illustrates the difference can be found in this question where the OP wants to do part of a LINQ to SQL query in SQL Server, bring the objects into managed code, and do the rest of the query in LINQ to Objects. To achieve this all he has to do is cast the IQueryable into an IEnumerable where he wants the switch to happen.