When going through a website migration you should always redirect old pages to the most relevant new page. For me, it doesn't matter whether that is an internal page or homepage. If you redirect the old homepage to the new homepage and the content is not relevant then you are going to have a high bounce rate as customers will not be seeing the right page for their search query.
It may be worth tailoring the design to show people that they are on a new website (sounds daft but users sometimes need the reassurance) and this way they are more likely to remain on the website even if things aren't exactly the same. It is probably a good time to go through your link profile to double check the keywords as well, make sure you have a mix of branded and non-branded keywords (which it sounds like you have already).