The major differences between JavaScript in IE and JavaScript in modern browsers (ex, Firefox) can be attributed to the same reasons behind the differences in CSS/(X)HTML cross-browser. Back in the day there was no de facto standard; IE/Netscape/Opera fought a turf war, implementing most of the specs, but also omitting some as well as making proprietary specs to gain advantages over each other. I could go on at length, but lets skip ahead to the release of IE8: JavaScript was avoided/scorned for years, and with the rise of FF and the contempt of webcomm, IE chose to focus mostly on advancing their CSS from IE6 on. And basically left DOM support behind. IE8's DOM support might as well be IE6's, which rolled out in 2001....so IE's DOM support is nearly a decade behind modern browsers.
If you are having JavaScript discrepancies particular to a layout engine, you're best bet is to attack it the same way we took on the CSS problems; Targeting that browser. DON'T USE BROWSER SNIFFING, use feature detection to sniff out your browser/it's level of DOM support.
JScript is not IE's own implementation of ECMAScript; JScript was IE's answer to Netscape's JavaScript, both of which came into existence before ECMAScript.
As far as type attributes on the script element, type="text/javascript" is the default standard (at least in HTML5), so you don't ever need a type attribute unless your script isn't JavaScript.
As far as IE not supporting innerHTML...innerHTML was invented by IE and is still today NOT a DOM standard. Other browsers have adopted it because it's useful, which is why you can use it cross-browser. As far as dynamically changing tables, MSDN says "because of the specific structure required by tables, the innerText and innerHTML properties of the table and tr objects are read-only." I don't know how much of that was true initially, but clearly the modern browsers have figured it out while dealing with the complexities of table-layout.
I highly recommend reading PPK on JavaScript
Jeremy Keith's DOM Scripting
Douglas Crockford's JavaScript: The Good Parts
and Christian Hellman's Beginning JavaScript with DOM Scripting and Ajax to get a strong grasp on JavaScript.
As far as Frameworks/Libraries are concerned, if you don't have a strong grasp on JavaScript yet, you should avoid them. 2 years ago I fell into the jQuery trap, and while I was able to pull off magnificent feats, I never learned a damn thing about coding JavaScript properly. In hindsight, jQuery is a wicked awesome DOM Toolkit, but my failure to learn proper closures, prototypical inheritance, etc., not only set my personal knowledge back, my work starting taking huge performance hits because I had no clue wtf I was doing.
JavaScript is the language of the browser; if you are client-side/front-end engineer it is of upmost importance that you command JavaScript.
Node.js is bringing JavaScript full tilt, I see immense strides taken daily in its development; Server-side JavaScript will be a standard in the very near future. I'm mentioning this to further emphasize just how important JavaScript is now and will be.
JavaScript is going to make more waves than Rails.
Happy Scripting!