The tele will definitely get you a fatter tone, but it will have slightly more of a 'jangle' too it- which can be good or bad depending on your preferences. I have a Walnut body Fender tele that sounds completely different than any of the alder or ash teles; much more midrangey and a very nice low end. It's probably the most Gibson-ish sounding Fender I've played. You may want to consider one of those or a rosewood body if you're looking for a more chunky sound. However, if I were forced to choose between keeping either my tele and my US Custom Shop strat, the strat would win any day.
In terms of versatility, the strat has more options, but there may be less usable ones soundwise (though a blender pot may help) when compared to the tele. The tele is a lot more versatile than it looks, it's been used by artists from Prince to Jeff Buckley to Jonny Greenwood (Radiohead) to Lou Reed to Johnny Marr (The Smiths) to The Dream Syndicate...