I'm not 100% sure on this, but my guess is that the "Slide to answer" appears when the phone is locked (screen is off) and the "Answer" or "Ignore" options come up when the phone is unlocked and you're using it (texting, browsing the web, etc.).
You can ignore either type of call by hitting the lock button once to silence the ring, or double-tapping the lock button to send it straight to voicemail (at least on AT&T).