As you can see below, the Vanguard REIT ETF fund hasn't been doing very well for the last 6 months, but the housing market seems to be improving every month. Why do we have this difference?
VNQ only holds ~16% residential REITs. The rest are industrial, office, retail (e.g. shopping malls), specialized (hotels perhaps?) etc.
Thus, VNQ isn't as correlated towards housing as you might have assumed just based on it being about "real estate."
Second of all, if by "housing" you mean that actual houses have gone up appreciably, then you ought to realize that residential REITs seldom hold actual houses. The residential units held tend primarily to be rental apartments. There is a relationship in prices, but not direct.
To round out something that @Chris W. Rea pointed out, the business that a REIT is in will be either A) Equity REIT... property management, B) mortgage REIT... lending, or C) hybrid REIT (both).
A very key point about why REITs broadly have been struggling lately, (and this would show up in the REIT indices/ETFs you've linked to,) is linked to the REIT business models.
For an Equity REIT, they borrow money at the going rate (let's say ~4.5% for commercial-scale loans), and use that to take out mortgages on physical properties. If a property rents for $15K per month, and they can take out a $1.8 million loan at $9,000 per month, then their business is around managing maintenance, operating expenses, and taxes on that $6,000 per month margin.
For a mortgage REIT, they borrow funds as a highly qualified borrower, (again let's say ~4.5%), and lend those funds back out at a higher rate. The basic concept is that if you borrow $10 million at 4.5% for 30 years, you need to pay it back at $50,668 per month. If you can lend it out reliably at 5%, you collect $53,682 per month... a handy $3,000 per month. The cheaper you can get money at (below 4.5%) and the higher you can lend it at (above 5%), the better your margin is.
The worry is that both REIT business models are very highly dependent on the cost of borrowing money. With the US Fed changing its bond-buying/QE/stimulus activity, the prevailing interest rates are likely to go up. While this has its benefits (inflation), it also will make it more expensive for these types of companies to do business.