Illustrator and web-browser SVGs use a common SVG technology but implement it differently - so what you can only use in both realms is a small subset of each. However, there could be custom code written for both Illustrator and a web browser that could adjust the SVG for proper usage. Here is an example with a business card, if you use the Submit button , it will download the SVG but you'll see that it's not an area-text, and things which are center-justified are actually left-justified but started at a position further right. But there's some extra art with special names which is used by the web script, it could also be used by some Illustrator script to turn it back into native elements. Past some very basic usage like this, it could be prohibitively hard to handle every browser svg aspect in Illustrator. Not as bad vice-versa though. SVG business card example:
* In this graphic the term "SVG Illustrator can handle" really means Illustrator's entire suite of art features, destructively exportable as an SVG.
So my advice is to export the SVG from Illustrator which looks similar to the one you wish to be using. Then, examine the differences between Illustrator's SVG export and what you are working with originally to determine if there are some changes which could be made to ensure the round-trip you are looking for. At the least you could learn which item is causing the hiccup and what web-svg implementations are a deal-breaker for Illustrator.