They do solve the same problem.
Let me start off with pro/con, then I'll move into technical differences.
git-annex
Pros:
- Supports multiple remotes that you can store the binaries.
- Can be used without support from hosting provider (for more details see here).
Cons:
- Windows support in beta, and has been for a long time
- Users need to learn separate commands for day-to-day work
- not supported by github and bitbucket
git-lfs
Pros:
- Supported by github, bitbucket and gitlab
- Most supported on all os's
- Easy to use.
- automated based on filters
Cons:
Technical
git-annex
git-annex works by creating a symlink in your repo that gets committed. The actual data gets stored into a separate backend (S3, rsync, and MANY others). It is written in haskell. Since it uses symlinks, windows users are forced to use annex in a much different manner, which makes the learning curve higher.
git-lfs
Pointer files are written. A git-lfs api is used to write the BLOBs to lfs. A special LFS server is required due to this. Git lfs uses filters so you only have to set up lfs once, and again when you want to specify which types of files you want to push to lfs.