We had our hot water tank replaced and now our water flows really slow out of the tap of our tub. Yet the flow is OK from the shower. We have the single handle faucet. It seems you have to pull the little lever for it to go to the shower a few times though for it to go to shower.

It is not uncommon for one or more faucet devices to go haywire immediately after a water heater change out, especially if the cold water line to the water is galvanized metal pipe and is quite old. while this scenario is the most common, it is not the only possibility. Sometimes it's just plain coincidence in the timing of events.
I would remove and replace the #1222 cartridge, inspect it for obvious debris from the heater replacement, briefly turn the hot water supply on to flush out any possible debris in the valve body itself, then reinstall the cartridge. Valve flushing should always be done before any cartridge gets reinstalled.
If the problem persists, the diverter on the tup spout would be my next focus. Again, debris from the heater replacement could (but far less likely) have affected its function. While not as easy to remove and exactly replace, by taking it off...then turning the water on, you will be able to observe the flow difference..With no difference the problem is in the valve and the cartridge should be replaced....with significant flow difference the, replace the tub spout. Be sure to take the tub spout with you, if it's the problem, when you take it to your plumbing supply store so it can be matched to a replacement. There are many tub spouts for many situations.
Search this site for Moen cartridge removal procedures. I have posted numerous solutions.