Illustrator still has to rasterize (bitmap) the image to export for the web, and has much less control.
There are **various** things you'll have to play with, and it will vary from font to font, and with the size of text. Fonts are inherently vectored art and render to the screen based on the information in them; a font with more control points in the characters will render differently than more simple fonts.
Another important factor is the resolution of the file you're working with (see sharpening below.)
You start when creating the type. There are settings for crisp. strong, and smooth for the text. The on-screen display is never really sufficient, you'll have to flatten and save to see the full effect of any of these. Any of the three may work out better for a given situation, there is no one setting.
Once rendered to your file, you want to flatten the file first, don't flatten the type to it's own layer. the reason for this is the way sharpening works (below) and if flattened to it's own layer, any sharpeing you do will hammer the edges of the type, sometimes even altering the shape of the characters or at the very least, create an extremely bitmapped edge.
Once flattened, you have to play with unsharp mask, but playing with it effectively requires an understanding of how it works. The sharpen and sharpen more settings are nothing more than unsharp mask presets.
Unsharp mask comes from a term in the printing industry where we used to make a copy of an image, then make a blurred copy of an image and superimpose the two negatives. Where the blurred (unsharp) edges meet the normal negative, it would create a fuzzy outline at only the edges which was used as a mask to overexpose those areas in the final. This created the contrast-edge effect described below, visually "sharper."
Sharpening is the effect of taking any areas of immediate contrast (edges) and widening/darkening the darker side and widening/lightening the lighter side. Really all you're doing is creating more contrast at the edges, but the visual effect is that of a sharper image due to the increased contrast at the edges. This is why when an image is over-sharpened objects appear to have an outline (amateurs. :-) )
So knowing this, when you go to sharpen your type, radius affects the width of the sharpening lines, tolerance affects the levels of difference between contrasting edges, and percent obviously works both of those to increase the contrast at edges. If your radius is too wide you will form outlines; if the levels are too high you will start to sharpen areas that are supposed to be flat, giving a granulated effect to the overall picture, and obviously percent needs to work between the balance of the two. Although there's no set numbers, a good starting point is anything under 1.0 for radius (0.6, 0.8 etc,) 4-6 for levels, and 70-90%. Note that high res files will react differently than 72 DPI files.
A final area of experimentation is with format. Jpg is a "lossy" format over which you don't have a great deal of control. What it does is creates a lookup table of the colors used in the images, and using anti-aliasing in combination with the level of compression you select, does all it's rendering by math. Sometimes it works fine, sometimes it's not a solution.
Indexed color (via .gif or .png) gives you the greatest control for type, but it comes at a small tradeoff of either file size or loss of tonal range. Indexed color has a set maximum 256 levels of color, so if you have gradients in your type they are likely to "step" and not be smooth. However if you can get your type "they way you like it" in Photoshop, it's more likely to stay that way when rendered as a .gif. File size is also important too; sometimes you can get a 2K image in .gif and it's fine, but as a .jpg it's 7K or more and degrades too much if you try to set the compression higher (quality lower.)
Last note, any sharpening you do, you have to do it before it converts to indexed color, sharpening has no effect on indexed color images.
So the only real answer to "sharp web text" is . . . there isn't just one. :-) It depends on what you're doing with the type.