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TTL......Transistor to Transistor logic... here as you said the voltae level for the logic is 5V for a 1 (high) or 0V for a zero (Low). Actually those voltages range have some tolerance, as you can see in the attached image (taken from this site I hoe they don't mind :cry: ) .
Schmitt Trigger inputs are different, they are set up so that they work around their hysteresis. Here you can find areally good explanation on how it works. In a simple phrase: The output on a Schmitt Trigger will only be high after passing a set reference votage A, and will turn low at a lower reference voltage B.
Schmitt Trigger is good when you have a noisy input and you want a clean digital output which could be TTL.