Learn English – the difference between “Come on home” and “Come home”

The main difference is one of formality and/or politeness.

Do come in is the most polite/formal (but not too formal to still be friendly) - the "do" has the sense of please feel free to.

Come on in is slightly less formal. It is often used when there is some sort of hesitation or delay on the part of the listener (e.g. "Come on in, the water's lovely!" - indicating that one brave soul has tested the water, while the others are waiting on the beach; or if someone arrives late and starts apologising on the doorstop "I'm sorry I'm late, we were stuck in traffic, ..." I might say "Don't worry at all! Come on in!")

In you come is the least formal, and comes across as fairly friendly. It is slightly more of a command than the others - I might use it to a small child or an animal, while at the same time lifting or otherwise assisting them to enter.

In my experinece you run into people and trouble:

I ran into Betty at the store today. I haven't seen her since college!
I ran into your co-worker Bob today.
Sorry I'm late; I ran into heavy traffic.

You come across objects:

I was cleaning up the kitchen and came across that phone bill you lost.
I was looking up (something) in the encylopedia and came across this interesting tidbit.

I'm not sure I'd use either when talking about a grammar error in a blog; I'd be more likely to say I found or noticed it. I don't know why that is.