It will most certainly be warmer in June. We go over spring break and last year's break landed April 15-19, the days we were at the park were perfect, low 80s, not very humid but prior to and right after it was about 90 and not as low of humidity. April is warm but not near the humidity of June and depending on where you're from, could feel like summer. We also had a few rouge coolish days in April too so it can vary but by end of May/early June, there's no more really rogue weather days, it's just summer. We mostly wore hoodies until mid morning in April but in June, you don't need a hoodie for early mornings/evenings. You may also have a little more chance of afternoon rain in June but we've found that's not as much of an issue as July-September.
We also go right after memorial day since our kids get out right before the holiday and crowds are not bad. By mid June they do creep up. You still may have lingering spring breakers in April but after Easter and that following week, they should dissipate so I'd venture to guess your April crowds might be a tad lower. I think it comes down to your priorities of cost saves, weather and what works best for you. If I could pick from the dates you gave me, I'd pick April over early June.