Spring is best. We dove with Deep Blue Utila this past March and were lucky enough to snorkel with 5 or 6 of them over two different days. Had the weather cooperated, we likely would've seen more. Steve, the owner told us that this past year they've seen at least one every month. He's on SB as DeepBlueDivers and is the best expert I know of. I'm sure he'll post here once he sees this.
There are 4 selected Research Weeks during the Feb-May timeframe. During those weeks experts are brought to the resort for lectures at night and they spend a higher percentage of their time looking for and documenting whalesharks. We were there during one of those weeks, they're priced slightly higher but they put on a spotting boat during those weeks so in addition to your two morning dives, you spend a couple hours daily looking for whale sharks as they're typically in deeper water. Their captain, Swin is legendary for spotting them and seemed to enjoy it every time he found one.
My buddy was able to get a clean shot of one and submitted it to their database, had it been the first spotting he would then have been notified via e-mail for life anytime it was spotted. We also (unsuccesfully) participated in a tissue sampling with a trained researcher, the best part of that was that the shark stayed on the surface for almost a minute while that was going on - as it was a little - well, angry lol...
And we saw another one feeding vertically which I was told is a rare experience. You can't
dive with them in Utila and the snorkels are of very short duration, usually once the shark sees anyone it dives deep. I was also fortunate enough once (dumb luck - I was late off the boat) to have one swim right past me, roll over on his side to eye me and then swim off. One kick and I could've touched it. It was about 15-16' long and I'll remember that forever.
I've adopted a whaleshark through the Shark Research Institute and get periodic updates on him. He lives off the coast of Tanzania and his name is Zahir. Links to stuff I've mentioned:
Deep Blue Utila swim with whale sharks at our all inclusive resort. Utila, Bay Islands, Honduras
Utila Whale Shark Research
Shark Research Institute- shark conservation, education, research