See Solving CF7 Javascript Conflicts.
Use Firebug or Chrome Dev Tools to examine Javascript conflicts in detail - see Use Firebug or Chrome Dev Tools.
Remember that a problem in one script can cause issues in another. For example, some jQuery errors can stop jQuery processing any more script, which can make the culprit hard to find. A jQuery error in one of your plugins or your themes javascript files can stop jQuery processing before it gets to the CF7 script.
You will need to be skilled at using Firebug to find what is causing your Javascript conflict and how to resolve it - see https://getfirebug.com/wiki/
Javascript Conflicts are a big deal. There are so many poorly written themes and plugins that cause Javascript conflicts with well written plugins like CF7. You have my sympathy.