Technically there's no difference, aida is a type of evenweave - meaning that it has the same number of threads in the warp (vertical) and weft (horizontal). Most cross stitch fabrics are evenweaves but there are a small number that are unevenweaves.
Aida is a blockweave, meaning that each square is made a group of threads - you can see that in this picture. Aida is sold in various counts indicating the number of blocks in an inch. The most common counts are 11ct, 14ct, 16ct, and 18ct. The higher the number, the smaller your stitches will be and the smaller your finished design will turn out.
This is 14ct aida in Soft Mocha from Wichelt.
And here's some stitching on aida were you can see how each stitch covers one square on the fabric.
Commonly known as evenweaves are the fabrics that aren't block weaves, the threads are not arranged in groups. Examples of evenweave fabrics are jobelan, lugana, brittney, davosa, murano, monaco. They are available in various counts, most commonly 20ct to 32ct, This denotes the number of threads per inch. Each stitch is usually formed over a grid of two threads square but more about that below.
This is Jobelan Lambswool 28ct from Wichelt.
And finally there is linen, also an evenweave but the difference here is the fibre used to make the fabric. Linen is made from flax, a natural plant fibre, and because of this it will have slubs or flaws in the fabric. Linen comes in various counts from 25ct to 50ct. Stitches are usually made over two threads on linen.
This is 28ct Natural Linen from Wichelt - you can clearly see the slubs in this picture.
Here's some stitching on an evenweave where you can see each stitch covers two threads vertically and two threads horizontally. This is known as stitching over two. You can see below the arrangement of the threads. Bring your needle up at 1, down at 2, up at 3 and down at 4 to complete the stitch.
The huge advantage to evenweave though is that you can stitch over one thread to make tiny stitches (sometimes known as petit point). The alphabet below is stitched over one thread.