Definitely it depends on what games you play. For esports and casual older games, a laptop is just fine. I used a laptop for everything (including a lot of gaming) until August, with a fairly crappy GT 635m graphics card. If you want to play AAA titles then a desktop is a better bet.
You don't have to spend a fortune on a gaming machine either. Best value for money is to get a refurbished office machine (like a Dell) with an i5 CPU and 8GB of RAM (second gen i5-2400 go for about $100-$150 on eBay), add a $150 GTX 1050Ti and you can play all the games at 1080p with decent settings. For less than $300.