tl;dr

ChronoUnit.YEARS.between( 
    LocalDate.of( 2010 , 1 , 1 ) , 
    LocalDate.now( ZoneId.of( "America/Montreal" ) ) 
)

Test for zero years elapsed, and one year elapsed.

ChronoUnit.YEARS.between( 
    LocalDate.of( 2010 , 1 , 1 )  , 
    LocalDate.of( 2010 , 6 , 1 ) 
)

0

ChronoUnit.YEARS.between( 
    LocalDate.of( 2010 , 1 , 1 )  , 
    LocalDate.of( 2011 , 1 , 1 ) 
)

1

See this code run at Ideone.com.

java.time

The old date-time classes really are bad, so bad that both Sun & Oracle agreed to supplant them with the java.time classes. If you do any significant work at all with date-time values, adding a library to your project is worthwhile. The Joda-Time library was highly successful and recommended, but is now in maintenance mode. The team advises migration to the java.time classes.

Much of the java.time functionality is back-ported to Java 6 & 7 in ThreeTen-Backport and further adapted to Android in ThreeTenABP (see How to use…).

LocalDate start = LocalDate.of( 2010 , 1 , 1 ) ;
LocalDate stop = LocalDate.now( ZoneId.of( "America/Montreal" ) );
long years = java.time.temporal.ChronoUnit.YEARS.between( start , stop );

Dump to console.

System.out.println( "start: " + start + " | stop: " + stop + " | years: " + years ) ;

start: 2010-01-01 | stop: 2016-09-06 | years: 6

About java.time

The java.time framework is built into Java 8 and later. These classes supplant the troublesome old legacy date-time classes such as java.util.Date, Calendar, & SimpleDateFormat.

The Joda-Time project, now in maintenance mode, advises migration to the java.time classes.

To learn more, see the Oracle Tutorial. And search Stack Overflow for many examples and explanations. Specification is JSR 310.

You may exchange java.time objects directly with your database. Use a JDBC driver compliant with JDBC 4.2 or later. No need for strings, no need for java.sql.* classes.

Where to obtain the java.time classes?

• Java SE 8, Java SE 9, Java SE 10, Java SE 11, and later - Part of the standard Java API with a bundled implementation.
• Java 9 adds some minor features and fixes.
• Java SE 6 and Java SE 7
• Most of the java.time functionality is back-ported to Java 6 & 7 in ThreeTen-Backport.
• Android
• Later versions of Android bundle implementations of the java.time classes.
• For earlier Android (<26), the ThreeTenABP project adapts ThreeTen-Backport (mentioned above). See How to use ThreeTenABP….

@Michael Borgwardt's answer actually does not work correctly in Android. Rounding errors exist. Example 19th to 21st May says 1 day because it casts 1.99 to 1. Use round before casting to int.

Fix

int diffInDays = (int)Math.round(( (newerDate.getTime() - olderDate.getTime()) 
                 / (1000 * 60 * 60 * 24) ))

Note that this works with UTC dates, so the difference may be a day off if you look at local dates. And getting it to work correctly with local dates requires a completely different approach due to daylight savings time.

Calendar dobDate; // Set this to date to check
Calendar today = Calendar.getInstance();
int curYear = today.get(Calendar.YEAR);
int curMonth = today.get(Calendar.MONTH);
int curDay = today.get(Calendar.DAY_OF_MONTH);

int year = dobDate.get(Calendar.YEAR);
int month = dobDate.get(Calendar.MONTH);
int day = dobDate.get(Calendar.DAY_OF_MONTH);

int age = curYear - year;
if (curMonth < month || (month == curMonth && curDay < day)) {
    age--;
}

This avoids looping and should be accurate to the day.