"Ich habe eigentlich keinen Hunger" - Actually I am not hungry but I am eating nonetheless (because the food tastes so good).

"Ich habe tatsaechlich keinen Hunger" - It seems you don't believe me, but I am really not hungry - so stop bothering me with the food you offer!

"tatsaechlich" emphasizes that a statement is true, often used if people didn't believe it in the first place.

"eigentlich" is a kind of diminisher: A statement is true in its core sense, but may not be useful or used nonetheless.

Hope this helps.

Du hast eigentlich Recht - If I think about it I believe you're right.
Du hast tatsächlich Recht - I didn't believe you, but now I know you're right.

"Tatsächlich" indicates a matter of fact ("Tatsache") while "eigentlich" is weaker and is used to express some uncertainty.

"Eigentlich" is also often used to express that something is not as it was planned, or ought to be:
Eigentlich solltest du schon im Bett sein.
Eigentlich wollten wir im Sommer an die See fahren.
Such sentences are expected to be followed by "aber ...". "Tatsächlich" is not possible in these cases.

OK, great. Thanks for the explanations. Much appreciated!

Und dann hat Heidegger noch uneigentlich aufgebracht und damit eine Flut von Flachwitzen ausgelöst. "Eigentlich wollte ich gar nichts essen." - "Und uneigentlich?"