Wire size won't necessarily let you weld hotter and deeper, but it will make your machine happier running hot. If you mostly weld pretty thick stuff and you find yourself generally running in the top third of your welder's settings, moving up in wire diameter will allow you to weld the same stuff at a lower setting. Since your welder doesn't really want to be welding in the very top of it's range, that's a good thing.
On the flip side, if you have been using your .030 and running mostly in the middle or bottom of your welder's range, going up to .035 will mean you're usually running your machine low, which is as bad as running it high all the time (in terms of consistant welding performance) If you are not running your machine at the top of it's range more often then not then you don't want to go up in wire size.
Other things to consider with wire size are heat and material deposition. Smaller wire means a smaller heat effected area (area that gets super hot), which effects your heat contraction after the weld and heat embrittlement. Also, smaller wire leads to lower material deposition, which is just how much metal you're laying down. That has a giant effect on how you weld. Welding the same joint will take longer, which can cost you time, but you get to go slower, which can make it easier to do a good job. It also means that you aren't carrying as much metal, which can make out of position welding easier. For me these factors are more important than weld settings when thinking about wire size, except if the settings for one diameter are currently way low or way high.
