Is the process easier in Illustrator-- Design to HTML/CSS? Or does it to build the UI elements easier?
Illustrator is better for the drawing stage of your UI than Photoshop but both are equal when it comes to checking spacing, color values, corner radiuses and so on when it's time to actually build that UI.
It's also a bit more straightforward to export an object drawn with Photoshop than it is to do the same with Illustrator.
Also, decent-to-good "Photoshoppers" are a dime a dozen because both businesses and schools have a weird love affair with Photoshop but decent-to-good "Illustrators" are harder to find. You'll have a hard time convincing your client to make its team of ADs and designers switch to Illustrator.
The initial files will be rasterised - so you won't be able to get the vector benefits of svg unless you export paths to illustrator - which is a bit shit and tedious.
PSD to * is such a bad, bad approach
You can use a vector drawing app for wireframes, for mapping out screen flows and interaction behaviour, and finally for creating the finished visuals for the app. Photoshop is only really suitable for the last of these tasks. There are some visual effects that only Photoshop can achieve, so you may find you have to use both. But I would always recommend starting with a vector drawing app.
If you don't want to use Illustrator (which has a clunky UI in my view) there are plenty of cheaper vector alternatives . Here are a few
Inkscape (open source and capable of professional results) http://www.inkscape.org/en/
Sketch (Mac only, praised for its ease-of-use) http://www.bohemiancoding.com/sketch/
Serif DrawPlus (Windows only, low-cost but very capable) http://www.serif.com/drawplus/ (no trial version, but they are selling the previous version for £10 (approx $16 usd): http://www.serif.com/promo/
Xara Photo and Graphic Designer (Windows only, again low-cost but very fast, capable and an excellent interface) http://www.xara.com/uk/photo-graphic-designer/
Edit: just to add, the apps above do not have anywhere near the amount of tutorials for learning as Illustrator (or the industry recognition), so factor that in when choosing a vector app if that's important to you.