Until recently, the train network had a monopoly on long-distance land-based transportation in France, so the long-distance bus network isn't as developed as in most European countries. Like everywhere, buses are as a rule a lot cheaper than trains (at least if you book in advance), but they're also a lot slower.
The fastest connections between Cannes and Paris are the high-speed train, and flying from Nice. Both take about 5 hours and both have a price structure where advance tickets can be cheap but last-minute tickets are very expensive. From Cannes to Nice airport, there's a hourly bus. You have two possibilities with the train: “regular” trains (check schedules and book on the SNCF website, the cheapest fare for travel tomorrow is 66€) and iDTGV (non-refundable fares, valid for one train only, doors closing 5 minutes before the train starts, fewer trains, but can be significantly cheaper depending on when you book).
Flights from Toulon and Marseille are also a possibility: slower than the high-speed train flying from Nice, but you might happen to find a cheaper ticket.
By bus, I think the only company with long-distance service on that stretch is Eurolines. There's a single bus per day, running overnight, which you can reach by regional train or bus. The bus is full for the whole upcoming week and the cheapest fare for the next week is more expensive than the train for tomorrow.
Another option is the overnight train from Cannes to Paris. It's more expensive at 106€ (as usual, it can be a lot cheaper if you book ahead) but you save a hotel night.
You can also take local trains. The SNCF website won't offer you slower connections, you can look up the schedules on the German railways website and book at the station (there are no advance discounts on local trains). But I think that for this trip, the total price is more than the high-speed train, except maybe for highest-load weekends. And it's a 12 to 13 hour trip. You might choose to stop along the way, for example in Marseille or Lyon, but even if you do so check high-speed connections as well.
If you're flexible with times, yet another option is car sharing. For two passengers, it's likely to be cheaper than all but the cheapest advance fares on the train or plane.