This battle has gone on for as long as anyone can remember.

Really, it began when Dave Mustaine was "let go" from the band.

I'm sure there has been an article on this already, but I would like to put my spin on the issue.

MetallicA:

Metallica debuted with a lot of contributions from Mustaine still remaining. Kill 'Em All, was the album title, and it was definitely a great place to start. After their strong debut, they progressed to Ride the Lightning, and then struck oil with Master of Puppets. You can tell right off the bat the first time you hear the intro/intro solo to their "balladish" type songs (Welcome Home, Fade to Black, One.) that they aren't the fast metalheads they seem to be. Those few songs show just how musically inclined the band is. And the instrumentals (Orion, Call of Ktulu, and To Live are to Die) speak for themselves in that 8-9 minute songs that don't seem repetitive, and don't have vocals are impossible to achieve without an extremely talented band. Although, I will give credit to Dave Mustaine for The Call of Ktulu.
But tragedy struck the band after Master of Puppets when Cliff Burton, beloved bassist for Metallica, was killed in a bus accident. And as a result, after 1988, when they released ...And Justice for All, there came a maturity over the group. Some say it was Bob Rock who softened them up, but it was, I think, a tribute album in some ways.
 

This is totally my theory behind all of this! ...And Justice for All was just like Metallica trying their hardest to forget about the loss of Cliff Burton (R.I.P.). And they couldn't hold back when '91 came around and let all of their feelings out with Metallica (The Black Album).
I believe that they saw the huge success of the album with sales over anything they had ever seen, and they tried to continue their style (softer) with Load and Reload.

 They really showed their "good side" again on the second disc. And S&M was cool, however didn't do much for me.
And then all hell broke loose with St. Anger. Great riffs, horrible sound. I liked their idea to record even the minor screw-ups and just go for it, but you have to admit, if they recorded the songs from St. Anger with the same setup as Ride the Lightning, Master of Puppets, or ...And Justice for All, there would be many more fans of the album.
 

Megadeth:

Megadeth has not had the same problems as Metallica. First of all, they have been able to stay consistent with their albums. Rust in Peace stands out as exceptional though.

Though Megadeth has gone through many drummers and guitarists, they are still able to keep a somewhat consistent musical style. And I almost put Megadeth and Metallica in a different category of metal because they sound sooo much different. Anyway, this consistency comes from their core of the band, of course, Dave Mustaine.
Killing is My Business...And Business is Good! Was Megadeth's debut with songs like Killing is My Business, and Mechanix (exactly like the Four Horsemen, but like 3x faster). The album got strong reviews. Then Peace Sells...But Who's Buying came out, and this album, like the first, got great reviews. This album is kind of where their musical ability got to shine through. With little things like the bass intro to Peace Sells...But Who's Buying. (Which later became the theme song for TV News) And this is where the band became famous, selling over 1 million copies of the album.
The hiring of Marty Friedman. (Incredible guitarist by the way.) He added a more melodic style to Megadeth which the band definitely needed. This is where Megadeth hit its climax...they released Rust in Peace. Accept ional with, again, the instrumental side of the spectrum especially highlighted

 Then Dave suffered from his injury while on a break from touring. Thus, the breakup of Megadeth occurred in 2002. However, Megadeth got back together just recently and released the album, The System Has Failed.
 

Hatfield and Mustaine-both great rhythm guitarists (even thought Dave does more lead work than James). Mustaine definitely goes for the speed approach when writing his riffs. James goes for the more epic kind of thing. Dave and James put fairly equal amounts of effort in to their lyrical work. Although it just goes to shpow that Dave puts heart and soul into his words, as James puts in $ signs while writing his.

Hammett and Friedman-both seem to give their band a melodic style. Friedman uses a sweeping approach to his solos, and sometimes, he will write almost random things. (E.g. the solo in middle of Five Magic’s.) Kirk has a different attack to melody with definitely speed (this is where the guitarists are related), but to contrast Friedman, Hammett uses more technicality in that he uses more tremolo, and other various "things to do on the guitar".
Ulrich and Menza-Menza definitely holds the band together (like any drummer), but he brings something extra to the band with drum solos at the beginning of songs. (E.g. Trust, Rust In Peace, and Skin O' My Teeth.) These little things like this really highlight the bands ability to "share" the spotlight. Whereas, Lars doesn't add anything to Metallica. He, basically, keeps the band in timing, and that's just about all. He adds little rolls in the middle of a song, but that's pretty much it.
Nested and Ellison-I'm using Nested instead of Burton for the simple reason that he has been with the band the longest. Bass guitar doesn't do much other than add deepness to a song. Unless of course you're in Iron Maiden, and the bass is just as important as the rhythm guitar. But I don't know much about Ellefson other than that he was the original and stayed with Mustaine through the lineup changes. Newsted is a good bass player, and I really like listening to his solo intro to My Friend of Misery.