Profile is a collection of preferences uses by conferences etc.
In the case of SIP a profile is also the name for the resulting SIP UA created by a particular profile.

Context is a narrowed down view of something, in the case of the dialplan a context is a set of extensions.  It's like having a dedicated set of extensions per distinct context name like parallel universes.  both the foo context and the bar context can have extension 2001.

A classic example of this is with the default dialing range of 1000-1019. This range appears in all three contexts that are defined in the default configuration:
conf/dialplan/public.xml in "public_extensions"
conf/dialplan/features.xml in "please_hold"
conf/dialplan/default.xml in "Local_Extension"

If a call comes in on the public context and gets routed to 1000 (or 1001, 1002,...1019) then it gets handled by "public_extensions" in public.xml. That extension transfers the call to 1000 in the features context (defined in features.xml) which executes a "please hold while I transfer your call" kind of operation and then transfers the call to 1000 in the default context (which is defined in default.xml).

Check out those three files. You'll see some cool stuff!
-MC