Television often takes creative license. Usually, only avid watchers will notice the discrepancies that arise, but sometimes it's just plain obvious:
Take Uncle Jesse, please. In the first season, his last name is Cochran. For the rest of the show, it's Katsopolis. (Ok, really important side note: I just went to Wikipedia to check the spellings on those, and found the absolute most amazing Wikipedia entry ever. So hysterical on so many levels. Please take a minute or two to read this while I dry my tears from laughing so hard. And to top it all off, at the very top it warns: "This article or section has multiple issues").
For a more recent example, what on earth is happening with the characters' ages on Friday Night Lights? I've already talked about the weirdness of a 30 year-old playing an 17-year old, but now the age-crisis is exploding: Matt is graduating high school but Julie is only a senior. So is Landry. Lyla and Tim turn out to be two years younger than Jason, yet they've all been best friends since "we played pee-wee football together" -- yeah, don't you remember Tim? They let the 1st graders play with the nursery school kids.
Do I even have to mention Aunt Viv from Fresh Prince? Um, yeah, that's two completely different people playing her. Hearkens back to the double-trouble Dicks on Bewitched. Similar story with Pacey's high school crush from Seasons 1 and 2, who makes a random appearance in the finale...but wait, it's a different actress! Why bring back the character if you can't even get the same girl to play the part?!
While these obvious changes are noticed by all, it's fun to watch for the subtle inconsistencies. Keep your eye out.