trace allows us to view how the code executed in details.

Debug allows line by line exection..

Trace and Debug - There are two main classes that deal with
tracing - Debug and Trace. They both work in a similar way -
the difference is that tracing from the Debug class only
works in builds that have the DEBUG symbol defined, whereas
tracing from the Trace class only works in builds that have
the TRACE symbol defined. Typically this means that you
should use System.Diagnostics.Trace.WriteLine for tracing
that you want to work in debug and release builds, and
System.Diagnostics.Debug.WriteLine for tracing that you want
to work only in debug builds. 

Tracing is actually the process of collecting information
about the program's execution. Debugging is the process of
finding & fixing errors in our program. Tracing is the
ability of an application to generate information about its
own execution. The idea is that subsequent analysis of this
information may help us understand why a part of the
application is not behaving as it should and allow
identification of the source of the error.