Yes, yes, yes, yes, yes
You must test on both platforms, otherwise you're leaving the door open to lots of conflicts. You say both platforms share a lot in common. Thus, IN THEORY should be the same for users. But... can you guarantee the coding on both platforms is flawless, that there are no mistakes and that potential platform issues have been covered? Can you guarantee it 100%? Without a doubt? If not... test!
As for times being tight, this is pretty common. Having time is a luxury most of the times we don't have. Blame it to bad planning, unrealistic expectations, further discoveries, cross-platform issues... well, it's pretty common. But there are not excuses for at least going to a college or mall and at least do some quick informal testing, just like Joel Tebbet said. No testing is way worse than poor testing.
The kind of problems you may have could be really weird. Coding issues, interaction issues, transition times, platform based expectations and many more.
If you want an example of what could happen, here's one big failure from yours truly:
we designed and started to built a great app, something we were really
proud of. When testing the main feature, it worked like a charm.
iPhone5 and 6, Galaxy5, Xperia... simply perfect. You could say
testing shown our work was flawless! Then we tested on a low end Motorola: we
burnt it. Literally: it started to heat and catch fire.
We tried to find out the
issue until we realized the computational needs exceeded almost any
low level smartphone. As simple as that. Just imagine if we launched
something tested on Android and iOS only that high end phones.
Happily, we discovered the problem at that stage.
Point is: not all issues and problems are those that you may reasonably expect. Testing is a tool to show you what you're missing (hence why you test with users and not stakeholders)
There are no excuses to avoid testing. In any case, you won't be able to do the perfect test you want to do, but something is better than nothing.