Well in C# 3.0 you can actually write:
public string Name {get; set;}
Which allows you to be proper and lazy.
Generally speaking, with properties, you get proper encapsulation. You have the choice to allow setting a value, or getting it, or both. Using a public member, you don't have that option.
It's probably one-part preference, and one-part how your team decides to handle quick and dirty class definitions, but I would say, use properties for get/sets.
To answer
Can anyone clearly articulate when you use an attribute and when to use a property in class design?
You shouldn't ever use a public attribute. You should always use a property instead. It's safer and more flexible. That said, people will be lazy, and just use a public member. However, with C# 3.0 you can use a more terse syntax to define properties, which should satisfy your inner laziness.
Simply type prop
and hit <tab>
to expedite the laziness in adding a property.
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