Bob
In these days of EASA, there are no 'Public Transport' or 'Private' CofAs for EASA aircraft (ie those that aren't orphaned machines like Chipmunks that come under the term 'Annex II'). Now we have a non-expiring CofA which may be 'normal' or 'aerobatic' for instance, revalidated by an ARC (Airworthiness review Certificate).
As Mike says, if you do carry out pilot allowed maintenance, the aircraft can then only be used for 'Private' use until recertified by an LAE or approved organisation (JAR145). JAR145 is required for AOC use (ie pleasure flying) but not for flying training/PPL hire.
What a pilot can do is listed in the ANO (see Mike's post). As for tyre changes, as SNS3G says, repairing a punctured tube on an aircraft is not quite the same as on a bicycle - probably cheaper and a lot less hassle in the long run to have it done by a 'qualified' engineer.