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That being said, we do not really know what makes hash functions resistant (see for instance this answer for a detailed discussion). With a lot of hand-waving, I could claim that SHA-1 is more robust than MD5 because it has more rounds and because the derivation of the 80 message words in SHA-1 is much more "mixing" than that of MD5 (in particular the 1-bit rotation, which, by the way, is the only difference between SHA-0 and SHA-1, and SHA-0 collisions have been produced).
For more of the same, look at SHA-256, which is much more "massive" (many more operations than SHA-1, yet with a similar structure), and currently unbroken. It is as if there was a minimal amount of operations for a hash function to be secure, for a given structure (but there I am moving my hands at stupendous speed, so don't believe that I said anything really scientific or profound). There are significant differences in their level of security against cryptanalytic attacks.
There are devastating collision attacks on MD5. These attacks mean that MD5 provides essentially no security against collisions: it is easy to find collisions in MD5.
In contrast, SHA1 appears to be much more secure. While there are some known attacks on SHA1, they are much less serious than the attacks on MD5. For this reason, SHA1 is a much better choice than MD5 in many settings.