While doing research in Massachusetts, I have found hand-written records of births, and then I find the Massachusetts Town and Vital Records, which are typed. Most of the dates match, but on some two dates are given and the abbreviation of G.R.1 follows the second date. Which date do I trust?
I'll answer the question from the question title first -- if a single source gives you two dates, yes, you need to record both of them, along with any other identifying information that tells you where the two dates came from. You'll need to analyze both and explain why they aren't the same when you write your proof statement, which is the last element of the five elements of the Genealogical Proof Standard. (Even if you don't plan to write up a formal proof statement like a professional genealogist might do, you may want to figure it out why the dates differ to satisfy yourself, and the GPS gives us the tools to do that.)
When you find any information, whether it is online or off-line, how do you evaluate what you have found? Elizabeth Shown Mills, the author of Evidence Explained, shows us how in her QuickLesson 17: The Evidence Analysis Process Map. (The Process Map has 9 elements arranged in three columns, and sometimes you'll see people refer to it by the nickname "the 3 x 3".)
Understanding how the records were created, and for what purpose, will help you decide which information is likely to be more accurate -- or whether neither record is correct. One single record does not mean "proof" of anything -- we have to weigh all the information we find and explain why it differs. See this answer for more comments about the five elements of the Genealogical Proof Standard and the 3 x 3. Unfortunately there is no simple answer to the question "Which one do I trust?" Original records can have mistakes in them, so you have to learn more about them and make your case for which one you believe to be the most correct.
I found a birth date for one of the German immigrants I'm studying in these records. The person, who was born in Germany, was indexed as being "born" in the Massachusetts town the record came from -- so use abstracts (the 'record detail' page, which Ancestry.com shows you before you click through to view the image) with caution. The entry I found in the printed town records also had a notation "G.R." -- so I asked:
Where did these records come from? What was the nature of the records?
Whenever you find a record, whether you are searching online or offline, see what information is available about the record collection itself -- the 'genealogy' of the record.
If you access the database Massachusetts, Town and Vital Records, 1620-1988 on Ancestry.com, the Source Information says:
Original data: Town and City Clerks of Massachusetts. Massachusetts Vital and Town Records. Provo, UT: Holbrook Research Institute (Jay and Delene Holbrook).
Further down on the page, we learn a bit more about how the records were collected:
Massachusetts boasts some of the oldest and most extensive records in the United States, and some documents in this database go back to the colony’s earliest days. They were made more readily accessible through the efforts of Jay and DeLene Holbrook. About 30 years ago, the Holbrooks began filming vital and town records in Massachusetts. To date, the Holbrooks have visited 315 of the state's 351 towns and cities and filmed their records collections.
The Holbrooks looked at the original records and transcribed them, so in addition to being a compiled set of records, this is a derivative work. Their notes were then typed up or typeset (or both) before being published. Any time a record is copied, no matter how diligent the copyists are, there is a possibility that someone could make a mistake, so it is always a a good idea figure out what the original records were and to look at them yourself, if you can get access.
Ancestry gives a list of the different kinds of records that were used to put together the collection. The Holbrooks coded the entries in their books so you can tell which source the information came from. "G.R.1" means the information came from a Gravestone Record -- someone's headstone. To find out which cemetery it is, use the browse function to go back to the start of the volume your record comes from, where you can find the list of codes.
It's a good idea to go back to the start of the volume for all sorts of records you find, whether online or offline -- don't just save or photocopy the single page you need, but look for lists of abbreviations, publication data, and other information that will help you evaluate what you just found.
The important things to consider here are that the birth date on a headstone is very far removed in time from the event it describes, and if the people who gave the dates to the headstone maker were younger than the deceased person, they can't have any personal knowledge about when the deceased was born. We also don't know when the stone was erected -- it could have been put up long after the person died.
Original Town Records
For the hand-written registers -- for example, from the collection Massachusetts, Town Clerk, Vital and Town Records, 1626-2001 on FamilySearch -- here are some things to look for that will help you evaluate the quality of the records:
1. Look for clues in the record header
This register for West Springfield births from 1857 has columns for the number of the entry, the date of birth, the name, and so on. Other registers from Western Massachusetts from the early 20th century have an extra column that says "date of entry".
2. Is the handwriting uniform?
Registers which are written all in the same handwriting may be copies of older original registers, especially if the handwriting is neat rather than sloppy. If a register had births from 1908 and all the dates of entries were from 1909, it's possible that something happened to the original register and the entries needed to be copied again.
3. Do multiple versions of the same records exist?
Depending on the requirements at the time, it's possible that cities may have kept their own records, while also submitting a copy of the information they collected to the state. If that's the case, then the local copy and the state's copy can differ. This is another reason that it helps to keep track of exactly what you saw and how you accessed it.
If you are using a collection from Family Search (whether you found it via Ancestry or Find My Past, or accessed it on FamilySearch directly), you can find out more information by going to the corresponding page on the Family Search Research Wiki (either by choosing the learn more link on the catalog entry page, or by searching the Wiki directly). Here's the article for Massachusetts, Town Clerk, Vital and Town Records (FamilySearch Historical Records) for the collection Massachusetts, Town Clerk, Vital and Town Records, 1626-2001 that the example came from.
(Be especially careful when researching Massachusetts marriage records, since it is possible to find marriage registrations in three separate towns -- one for the groom's place of residence, one for the bride's place of residence, and one for the locality where the event took place, if it is different from the other two towns.)
Meet the Holbrooks
You can find out more information about the Holbrooks in this news article Former Oxford couple create database of Massachusetts history that was written in 2012 when their research was published online by Ancestry.com.