First off, both genres have been strongly influenced by blues (so has jazz, for that matter). However, country also has a very strong Celtic influence. Country is also more closely related to jazz in the way it's played where rock is more closely related to jazz (but both country and rock do have blues and jazz influences). Rock was also influenced by country music. To hear that influence, listen to Chet Atkins then listen to some of the Beatles' music. George Harrison's guitar playing was influenced by Chet Atkins' playing quite a bit!
Secondly, rock music in general tends to be more open to more complicated chord progressions. Of course, that's not always true but, for the most part, it is. A lot of country artists tend to stick to nice simple chord progressions. Don't take that the wrong way though--rock artists still often use very simple chord progressions and country artists still often use more complicated chord progressions. Also, in rock, melodies tend to be more complicated than country...once again, a country song still can have a complicated melody. Rock also tends to use a wider variety of scales than country. For instance, rock guitarists will often use pentatonics scales and major scales which are also common in country. However, rock guitarists will also use harmonic minor scales and much more. The only country artist I can honestly think of who has uses harmonic minor scales is Brad Paisley and it was in his surf rock instrumental "Turf's Up."
Another of the main differences between rock and country are the instrumentals most often used. In rock, there's mainly just guitar, bass and drums. Of course, sitar is also used quite a bit (thanks to the Beatles introducing it into rock) by certain artists and banjo has also been used on occasion in rock. Piano and harmonica are also quite common and even more so than sitar and banjo. Some rock artists, including Led Zeppelin and Cinderella, have also used mandolin which is more often associated with country music but it an Italian instrument (it sounds great in Italian music too!). Now, in country, you have more main instruments. Some of those are guitar, bass, drums, pedal steel, fiddle and banjo. Of course, piano, harmonica, mandolin and a lot of other instruments can be used. Honestly, an instrument is not what makes a certain song (or artist) qualify as a genre. In reality, you could take a song like "Okie from Muskogee" by Merle Haggard and add a sitar and it could still stay a country song (if arranged right). And, in rock, well, look up country guitar virtuoso Brad Paisley's rock instrumental "Cliffs of Rock City" to see how fiddle could be used (that song's a tribute to rock guitarist Eric Johnson). Also, look up the rock band Warrant and their song "Uncle Tom's Cabin" for an example of how banjo could be used in a rock song.
In all honesty, country and rock really are more closely related than people tend to think. Of course, with modern country, you usually get artists who sound more like pop while less modern country artists tend to sound more like rock and even less modern country artists actually sound like country.
Some great rock artists to check out:
Badfinger
Tesla
Scorpions
Cinderella
Jackyl [watch out for the bad language with this band though]
The Beatles
Led Zeppelin
Mountain
Lynyrd Skynyrd
Allman Brothers Band
Molly Hatchet
Whitesnake
Chuck Berry
Jerry Lee Lewis
Eric Johnson
Warrant
White Lion
Some great country artists to check out:
Brad Paisley [some of his instrumentals are not country but, for the most part, he stays true to country while sometimes incorporating non-country influences into his guitar playing]
Alan Jackson
Jerry Reed
Hank Williams, Sr.
Chet Atkins
Buck Owens
Katrina Elam [more on the pop side, but a great vocalist nonetheless]
George Strait
Cross Canadian Raweed [labeled as country, but should be labeled as rock...they're a great band though]
Ashton Shepherd
Jamey Johnson
Roy Clark
Redd Volkaert
Merle Haggard
Patsy Cline
Waylon Jennings