FWIW I have both; desktop does (or rather did) heavy lifting - i.e. stuff that needed a decent size monitor, decent processing power lots of typing so proper keyboard. Laptop is small and used around the house / mainly for web browsing in bed.
In the desktop's favour it's cheaper for a given spec, far more standardised / easy to upgrade / repair and capable of storing a lot more drives / data. I bought mine new in 2009 (1st gen. i7, highish-end at the time) and it's still going strong despite being old-hat; having in this time had a new PSU (old one died), graphics card (old one died), SSD (far faster than an HDD), and a RAM upgrade.
In the laptop's favour it's obviously a lot more portable so can be used around / outside the house, while it doesn't require its own dedicated space, takes up a lot less room and is easier to tuck away. If you only want one machine but some of the versatility of the desktop, you can run the laptop with an external monitor, keyboard and mouse; made easier by using a laptop dock.
Sadly it seems that laptops have a much shorter lifespan than desktop as they have a harder life and are much more difficult / limited to upgrade. While not immediately obvious from viewing stuff online, consumer-grade latops are basically rubbish; their lower price reflected in a nasty user interface (mouse and keyboard), poor screens (colour rendition, viewing angles) along with generally shonky build quality, limited lifespan and poor upgradability. Also bear in mind that many laptops (esp. smaller ones) now come with RAM soldered directly to the motherboard - meaning what you get initially is it / you can't upgrade to more.
If you have the space and the application permits a desktop is the way to go; decent used ones can be had for very little money and not everything has to be some massive tower - there are a lot of nice small / ultra small form factor machines about. If going for either laptop or desktop I'd recommend Dell and Lenovo products.
Alternatively as others have said there might be scope in upgrading your current machine - do you know what processor it has / how much RAM is installed? Might just be that fitting a solid state drive and some more RAM will sort it out for not a lot of money
In the desktop's favour it's cheaper for a given spec, far more standardised / easy to upgrade / repair and capable of storing a lot more drives / data. I bought mine new in 2009 (1st gen. i7, highish-end at the time) and it's still going strong despite being old-hat; having in this time had a new PSU (old one died), graphics card (old one died), SSD (far faster than an HDD), and a RAM upgrade.
In the laptop's favour it's obviously a lot more portable so can be used around / outside the house, while it doesn't require its own dedicated space, takes up a lot less room and is easier to tuck away. If you only want one machine but some of the versatility of the desktop, you can run the laptop with an external monitor, keyboard and mouse; made easier by using a laptop dock.
Sadly it seems that laptops have a much shorter lifespan than desktop as they have a harder life and are much more difficult / limited to upgrade. While not immediately obvious from viewing stuff online, consumer-grade latops are basically rubbish; their lower price reflected in a nasty user interface (mouse and keyboard), poor screens (colour rendition, viewing angles) along with generally shonky build quality, limited lifespan and poor upgradability. Also bear in mind that many laptops (esp. smaller ones) now come with RAM soldered directly to the motherboard - meaning what you get initially is it / you can't upgrade to more.
If you have the space and the application permits a desktop is the way to go; decent used ones can be had for very little money and not everything has to be some massive tower - there are a lot of nice small / ultra small form factor machines about. If going for either laptop or desktop I'd recommend Dell and Lenovo products.
Alternatively as others have said there might be scope in upgrading your current machine - do you know what processor it has / how much RAM is installed? Might just be that fitting a solid state drive and some more RAM will sort it out for not a lot of money
