What’s the difference between punk rock and alternative rock?

I know that alternative rock incorporated several elements from old-fashioned punk rock but other than that I still don’t see the difference between the two genres that much. Help clarify it for me, please.

Yikes, these answers are terrible! Punk music is hard to describe, because it is less of a music genre and more of a movement. Real punk focuses on DIY ethos, that is "do-it-yourself" and also think-for-yourself approach to everything. It is about independence, and rebellion against the inherent oppression of a forced social structure. Second, it does has a certain music style (the cliche "three chords and a drumbeat") but that is besides the point. Notable punk bands include the following:
Crass: http://www.youtube.com/watch?v=Aen9I_JmdtI
Zounds: http://www.youtube.com/watch?v=m8w14IaS8Us
X: http://www.youtube.com/watch?v=wRWunSUmEm4
Discharge: http://www.youtube.com/watch?v=ZhoJj022nQk
Rudimentary Peni: http://www.youtube.com/watch?v=PsuOeFiKoXo
Alternative is a little complicated, because it can be broken up into two parts:
(1) First, it as a condition that applies to music that is outside of mainstream (or, alternative to mainstream). However, bands that were once considered Alternative began to make it big (such as The Smiths and The Cure) and consequently made this term near obsolete. An Alternative band can not be mainstream, because (by contradiction) it would no longer be alternative. Therefore, plenty of punk bands can be considered Alternative, because no (real) punk band is mainstream.
(2) The second definition of Alternative has to do with sound. The ‘sound’ began around the time of post-punk, when bands were taking punk ethic and applying to new, artistic, and experimental music. Sooner or later, these bands could not be placed into any genre, so they were called "Alternative" from there on.
Notable Alternative rock bands include the following:
The Jesus and Mary Chain: http://www.youtube.com/watch?v=noeTCf8PLuc
Sonic Youth: http://www.youtube.com/watch?v=xRi2epOS5SA
Joy Division: http://www.youtube.com/watch?v=PldpBJEn4vQ
REM: http://www.youtube.com/watch?v=KA57Pafq_NU
Pixies: http://www.youtube.com/watch?v=v3r-xMt9JmM
In conclusion, the difference between punk rock and Alternative rock depends on which definition of Alternative implied. Punk is (conditionally) an Alternative band because it is non-mainstream, but is not in the second, sound-relevant definition.
Hope that helps.