What is difference between "PowerSlide" and "Drift"?

This video shows the difference between those two in GTA:

...or Forza:

In short, a drift starts before the apex of a corner while a powerslide starts behind it.

• Drift = oversteer on your way in
• Powerslide = oversteer on your way out

And for those who want more insight: A real life explanation of Drifting vs. Powerslides:

While dly's answer covers the more technical definition of the terms, I thought I'd address some points specifically about CTR, as the concept is slightly different in that game.

In Crash Team Racing, 'Powersliding' refers to the entire 'slide' action, from the moment you land your kart and begin the slide, until the moment you release the bunny-hop/drift button (or spin-out). Note that in this definition the slide is not about 'oversteering' or putting your foot down to get the back wheels to spin - you aren't entering a slide because of a loss of traction, you're physically bunny-hopping the kart into a sideways alignment (relative to the direction of travel):

_{Animated Gif from the NeoGAF forums}

This is described in the in-game Adventure Mode hints as such:

Power Sliding Hint

You can go faster through turns by power sliding. To access the power slide, press and hold the R1 button while turning hard.
_{- Aku Aku and Uka Uka Hints}

And on Wikipedia:

A crucial maneuver in Crash Team Racing is the power slide; the player executes the slide by holding down one of the shoulder buttons to perform a hop, and steering before the kart lands.
_{Crash Team Racing - Wikipedia}

However, note that despite this in-game use, the terms 'powerslide' and 'drift' are used pretty much interchangeably by the community, sometimes even combining to two. NeoGAF User RK128 describes the powerslide mechanic as such:

The power drifting is also very different from Mario Kart, where instead of drifting until you get a colored spark to tell you 'Let go to move fast' you rather use one trigger to drift and the opposite trigger when the drift bar is green. You can do this up to three times and it makes drifting so rewarding in CTR.
_{Crash Bandicoot Retrospective - Crash Team Racing - NeoGAF Forums}

And this PlayStation LifeStyle article uses both terms but leans toward 'drifting' over 'powerslide':

Powerslides and drifting also took a notable upgrade in CTR. Drifting around turns has almost always been a simple way to get a small boost since Mario Kart 64 in 1996. However, Crash Team Racing was the first to explore the concept of the powered-up boost. Drifting around corners built up a boost meter and toggling a boost just before a catastrophic spin-out would allow players to score a better version of the regular boost, making drifting a special exercise timing and precision.
_{"It's Time for Crash Team Racing: Remastered - T.J. Denzer, August 22, 2017}

Note that the 'boost' mechanic is a separate action to a Powerslide. It's known as Turbo Boost in game, or sometimes 'Drift Boost', 'Slide Boost' or simply 'boosting' by the community. It's worth mentioning but it's a large enough topic all to itself so let's just focus on the slide part of the mechanic here.

So to directly answer your question:

Is there a difference?

I can't speak for Burnout as I've never played it, and I'm sure more simulation-racing games like Gran Turismo probably differentiate between the terms in the manner described by dly's answer, but for CTR specifically? There isn't really a difference. You should be ok to use them interchangeably in this context.