Both the size and the shape are different. 垃圾桶is a bigger cylinder while 垃圾筒is a smaller cylinder. And 垃圾箱often is the big cubic container.To be honest，we hardly ever distinguish this three words,your question drives many Chinese crazy.

They have different sizes, generally speaking, 垃圾箱 is the biggest, then is 垃圾桶, and 垃圾筒 is the smallest.

Usually we say 垃圾桶.,We don't say 垃圾筒.because 筒is too small and thin, and it can` t contain much.Maybe 垃圾筒called by you is 纸篓，generally speaking, 垃圾箱 is the biggest, then is 垃圾桶, and 垃圾筒 is the smalles, in mainland of China, inhabitants carry 垃圾 to 垃圾桶， workers carry 垃圾桶 to 垃圾箱，trucks carry 垃圾箱to outside city.垃圾子車.is said in Taiwan

也许各地区对装垃圾的东西的称呼不同，但是，现代汉语的角度来说，这三种称呼对应的是不同形状的装垃圾的容器。与大小无关。 比如：城市中有一种公共的垃圾桶，非常的大，要比一般的垃圾箱大的多，下面的轮子，垃圾车有专门的挂钩把这种垃圾桶举起来将垃圾倒入垃圾车中。几个都未必举的动它。而城市中到处安放的垃圾箱和它比连三分之一都到不了，谈什么bigest。 所以：三个名称对应的只不过是三种形状不同的用于装垃圾的容器罢了。桶状、筒状和箱状。 字纸篓，也是垃圾篓，也有叫垃圾筐的。只不过是因为它往往用于办公室之类的地方，装的大部分情况下是废纸之类，所以才叫字纸篓或纸篓。因为这种容器多半都是用塑料做成的筐子状的容器，因此许多人也叫它垃圾筐。 总之，这些名称都是从形状上来命名的，与大小无关。每一种都可以做得很大，也可以做得很少。

Usually we say 垃圾桶. We don't say 垃圾筒. I hardly ever hear people say 垃圾箱. We say 垃圾子車.