Can someone answer on my dilemma which method to use for connecting Android device to mySQL or Postgresql?
I can do it in both ways without any erro
Besides all things Craig Ringer said, which I completely agree, JDBC has another problem: it will force to expose your database to the world. If you want android devices to access it, you will need to provide your app with database credentials, and the database will have to have public access.
Using a WebService or RESTful API is clearly the way to go to make your application secure.
I can think of a few reasons
Web services can provide additional features on top of the JDBC connection like authentication / quality of service / authorization / conditional GET requests / error handling etc. JDBC cannot do any of these.
Another option would be to use a database sync tool like SymmetricDS.
This would let you have say a Postgres database on your server, and a SQLite database on your tablet.
SymmetricDS would synchronize the databases over HTTP, when a connection is available. You don't have to sync the whole db of course, just the relevant parts.
(I am not affiliated with SymmetricDS)
You think it's simpler and faster to do it with JDBC because you aren't considering the real world operating environment of phones and portable devices. They often have flakey connectivity through buggy traffic rewriting proxies and insane firewalls. They're typically using a network transport layer that has high and variable packet loss rates and latencies that vary over many orders of magnitude in short spans of time. TCP really isn't great in this environment and particularly struggles with long lived connections.
The key benefit of a web service is that it:
Has short-lived connections with minimal state, so it's easy to get back to where you were when the device switches WiFi networks, to/from cellular, loses connectivity briefly, etc; and
Can pass through all but the most awful and draconian web proxies
You will routinely encounter problems with a direct JDBC connection. One challenge is reliably timing out dead connections, re-establishing sessions and releasing locks held by the old session (as the server may not decide it's dead at the same time the client does). Another is packet loss causing very slow operations, long-running database transactions, and consequent problems with lock durations and transactional cleanup tasks. You'll also meet every variety of insane and broken proxy and firewall under the sun - proxies that support CONNECT
but then turn out to assume all traffic is HTTPs and mangle it if it isn't; firewalls with buggy stateful connection tracking that cause connections to fail or go to a half-open zombie state; every NAT problem you can imagine; carriers "helpfully" generating TCP ACKs to reduce latency, never mind the problems that causes with packet loss discovery and window sizing; wacky port blocking; etc.
Because everyone uses HTTP, you can expect that to work - at least, vastly more often than anything else does. This is particularly true now that common websites use REST+JSON communication style even in mobile web apps.
You can also write your web service calls to be idempotent using unique request tokens. That lets your app re-send modification requests without fear that it'll perform an action against the database twice. See idempotence and definining idempotence.
Seriously, JDBC from a mobile device might look like a good idea now - but the only way I'd even consider it would be if the mobile devices were all on a single high-reliably WiFi network under my direct control. Even then I'd avoid it for reasons of database performance management if I possibly could. You can use something like PgBouncer to pool connections among many devices at the server side so connection pooling isn't a big problem, but cleanup of lost and abandoned connections is, as is the tcp keepalive traffic required to make it work and the long stalled transactions from abandoned connections.