One of the big differences that stuck out to everyone is the look. With the 80's it was all about glam and big hair as seen in the photo of Motley Crue shown below.
In the 80's it was all about pleasing the girls and part of the way to do that was to make yourself look pretty. Before these bands came along girls were not really into metal before the 80's but with this era of glam going on it gave girls a way to find it appealing. Throughout the 80's bands dressed in spandex, carried around 2 cans of hair spray for big hair (well maybe only one!), and wore lots of make up. The make up became less of a thing later in the 80's but the type of songs they did stayed the same through the decade.
With grunge it was the other way around. These bands not only did not wear makeup but they didn't really look like they kept themselves together much at all. With their tee shirts, Flannel shirts, and torn up jeans they really never looked clean as seen with this picture of Nirvana.
The grunge seen was really angry and rebellious compared to the bands of the 80's. These bands were seen as misfits and they liked it that way. They didn't want to come off as happy souls they wanted people to see them as tortured souls not wanting to conform to society.
When it comes to music it was simplicity vs complex playing. In the 80's the musicians wanted to be as good at their instruments as they could. So many great guitar players came out of the 80's such as Eddie Van Halen, Yngwie Malmsteen, Warren DeMartini, George Lynch, and Steve Vai. One of the staples of 80's metal was to bust out killer guitar solos by using speed and tricks as seen in the videos below.
Another staple of the 80's metal music were power ballads. As I've said, this was a time of trying to appeal to the girls and one of ways to do that was through a ballad.
Lyrically they focused on mainly sex and love with some few exceptions. Even though a ballad is good way to show the emotion and human side of a rocker these bands did it so much to the point you couldn't take it seriously, which played a big part in the rise of grunge in the 90's.
Grunge music more closely resembled punk then anything else. The musicians knew very few chords, didn't master scales, and had a reputation of being low level players. If you ever go to see a grunge band expecting an Eddie Van Halen solo you will be very disappointed because these guys didn't believe in that kind of playing, and didn't have do pull it off anyhow.
Now where these bands exceeded was their lyric writing. Unlike how the 80's were all about sex these guys tackled a person's darkest hour. With that it felt like they were being real and authentic which was a big plus with a lot of people.
Also the concerts were different as well. With the 80's it was all about stadiums and playing in front of 30,000 people. If you were a band in the 80's nothing was cooler then playing in front of a ton of people and putting on a big show.
If you were a grunge band that was a terrible thing. Grunge bands would do stadiums and large festivals because they had to promote themselves but the majority of the time they preferred smaller venues with less people
Both genres had their ups and downs. The 80's seemed to produce the better musicians, and their playing was an influence for many current players. However, their song lyrics had a narrow focus and in the end they lacked substance and authenticity. In this respect the 90's seemed to produce much better lyricists than the 80's.
Another problem these bands had was that there were so many bands that it became sickening and people wanted something different. Even though the genre hit rock bottom in the 90's, many of the bands survived and still have a big influence. They influenced bands all over the world in places like Japan where bands like XJapan came along and through that Visual Kei was born. Visual Kei was heavily inspired by the 80's hair bands and it is a very successful music genre in Japan. Also many of those bands that survived the 90's still have a big appeal all over the world.
With grunge you got the sense of authenticity, and felt like the musicians meant what they were singing. They were easier to relate to personally. They were musicians that spoke from the heart and would play a big part in how to some people thought they are a good inspiration because they show that you didn't need to be great to make good music. However that was also one of their problems as well. Because they were not the greatest musicians in the world they created a void where people wanted to hear more advanced music and it would allow bands like Dream Theater and TOOL to gain fame. One of its biggest problems was that is was very short lived in popularity, Once Kurt Cobain died the genre slowly died with him. Pearl Jam took a long break, Sound Garden would break up in 1997 and Alice In Chains took a long break with Layne Staley's death in 2002. However, many of the bands have since reunited and like the 80's still remain popular through out the world.
Personally, I believe that we can learn things from both genres and I will encourage people to listen and find to good things in both and stop having a rivalry between them.