Well, they aren't really intended for the same type of service. BART is rather unusual by combining what is in some places almost commuter-rail station spacing with full grade separation ($$$), but it's still best when running more high-demand, high-frequency service. I wouldn't expect local SJ-SF BART service to be a lot faster than local Caltrain - faster acceleration would help a bit, but at that stop spacing it's not a huge factor.
In the long term there would definitely be utility for round-the-bay BART service, but right now Caltrain provides way better nortbound service than San Jose-East Bay connections, so that extension does make more sense for having more priority. With any luck, Caltrain electrification driven by the HSR project might happen before Santa Clara gets around to funding BART to Mountain View, further reducing the need.
Would electrifying Caltrain make it run faster? I'm not very well-versed in trains.
The other issues with Caltrain that I have include: The last Northbound train on Sundays being at 9pm and the last Northbound train on other days being around 11pm. Also, the fact that there are regions of time when they go every hour. Unfortunately, I doubt electrifying the train will really affect either of those.
It would make it a little faster since electric trains can accelerate out of stations a little faster. It's not a major difference at Caltrain's station spacing though.
The other issues you mention have little to do with diesel/electric Caltrain/BART differences. Caltrain could easily run late trains if its funding allowed, and in fact might be able to run later than BART, which begins shutting down at midnight to do maintenance on its raised and tunnelled track. Mid-day and weekend frequency is again a function of funding/chosen service level - more is clearly possible since there's 5 trains an hour during weekday rush hours.
One difference that might arise from electrification is support for electric multiple unit rolling stock. This is a setup where rather than being pulled/pushed by one big locomotive, each passenger car has smaller motors and propels itself. (BART cars are like this, but this can also be done with more commuter-level service.) This way the costs for three shorter trains an hour vs one long train an hour are closer than in diesel, which has the big fixed cost of the heavy locomotive, and it becomes more feasible to operate same capacity but more frequent service during times of less demand.
But again, this is also possible with diesel (diesel multiple unit) and has more to do with type of service Caltrain chooses/is asked to operate rather than electrification or dedicated right of way.
The real difference stems from funding models and goals of the service. BART is treated as near-metro (and in the central parts, is a metro) where moving people fast and conveniently is a goal. Minimum service is kept at 20 minutes even in outlying areas at 11 pm on a Sunday, not because there's many people taking it, but because that's the standard. Caltrain is primarily a commuter service through wealthy areas to make fairly rich people's commutes a little easier. People taking trains at 9 pm are not a particular priority.
Thanks, that's quite enlightening. I hadn't considered how much electrification could make things more efficient and thus easier to make more convenient.
In the long term there would definitely be utility for round-the-bay BART service, but right now Caltrain provides way better nortbound service than San Jose-East Bay connections, so that extension does make more sense for having more priority. With any luck, Caltrain electrification driven by the HSR project might happen before Santa Clara gets around to funding BART to Mountain View, further reducing the need.