Contemporaneous is "at about the same time", with the duration of "about" varying with context.  If you're discussing geology or evolution, "about" may be thousands of years, while if you're discussing baseball players maybe a few years.

Simultaneous is at exactly the same instant.

Contemporaneous refers to the duration of an event/lifetime.

Simultaneous is at exactly the same instant

<<A good friend will bail you out of jail, but a true friend
will be sitting beside you saying " Damn that was fun!" - Unknown>>

sharing a period of time vs. happening at the same point in time

 

Another difference is that I've spoken and written the word "simultaneous" many times, whereas I have never spoken the word "contemporaneous", and have only written (typed) it just this once.

Cheers,
CanuckMiner
 

Contemporaneous is a legal term meaning the same as people have said above.

look at the roots of the word
Contemporaneous and contemporary have the same latin root, however Contemporaneous is more often applied to things rather than people.  The definition of contemporary does allow things to occur simultaneously.  According to Oxford, simultaneous is not defined as "exactly" the same time - the words are synonymous. (look up Synchronism in Roget's)

contemporary
/kntemprri/

  • adjective 1 living, occurring, or originating at the same time. 2 belonging to or occurring in the present. 3 modern in style or design.

  • noun (pl. contemporaries) 1 a person or thing existing at the same time as another. 2 a person of roughly the same age as another.

  — ORIGIN Latin contemporarius, from tempus 'time'.

simultaneous
/simmltayniss/

  • adjective occurring, operating, or done at the same time.

  — DERIVATIVES simultaneity /simmltnayiti/ noun simultaneously adverb simultaneousness noun.

  — ORIGIN from Latin simul 'at the same time'.